The present invention relates to ellipsometers and polarimeters and the like, and more particularly is a Spectroscopic Rotating Compensator Material System Investigation System including a Pseudo-achromatic Compensator, and Photo Array for simultaneously detecting a Multiplicity of Wavelengths, which Spectroscopic Rotating Compensator Material System Investigation System is calibrated by a Mathematical Regression based technique involving, where beneficial and desired, Parameterization of Calibration Parameters. The present invention provides a preferred fast axes offset, dual zero-order, or effective zero-order, or combination zero-order and effective zero-order waveplate compensator system, alternative use of D.C. and A.C and combination A.C. and D.C. data normalizing bases in various calibration steps, as well as use of various material systems during calibration data acquisition. The present invention system can be realized utilizing off-the-shelf, non-ideal, waveplates.
Ellipsometry is a well known means by which to monitor material systems. In brief, a polarized beam of electromagnetic radiation of one or more wavelengths is caused to impinge upon a material system along one or more angles of incidence and then interact with a material system. Beams of electromagnetic radiation can be considered as comprised of two orthogonal components, (ie. xe2x80x9cPxe2x80x9d and xe2x80x9cSxe2x80x9d), where xe2x80x9cPxe2x80x9d identifies a plane which contains both an incident beam of electromagnetic radiation, and a normal to an investigated surface of a material system being investigated, and where xe2x80x9cSxe2x80x9d identifies a plane perpendicular to the xe2x80x9cPxe2x80x9d plane and parallel to said surface of said material system. A change in polarization state in a polarized beam of electromagnetic radiation caused by said interaction with a material system, is representative of properties of said material system. (Note Polarization State basically refers to a magnitude of a ratio of orthogonal component magnitudes in a polarized beam of electromagnetic radiation, and a phase angle therebetween.) Generally two well known angles, (PSI and DELTA), which characterize a material system at a given Angle-of-Incidence, are determined by analysis of data which represents change in polarization state.
Continuing, Ellipsometer Systems generally include a source of a beam of electromagnetic radiation, a Polarizer, which serves to impose a linear state of polarization on a beam of electromagnetic radiation, a Stage for supporting a sample system, and an Analyzer which serves to select a polarization state in a beam of electromagnetic radiation after it has interacted with a material system and pass it to a Detector System for analysis therein. As well, one or more Compensator(s) can be present and serve to affect a phase angle between orthogonal components of a polarized beam of electromagnetic radiation.
A number of types of ellipsometer systems exist, such as those which include rotating elements and those which include modulation elements. Those including rotating elements include Rotating Polarizer (RP), Rotating Analyzer (RA) and Rotating Compensator (RC). The present invention is, in its primary embodiment, a Rotating Compensator Ellipsometer System. It is noted that Rotating Compensator Ellipsometer Systems do not demonstrate xe2x80x9cDead-Spotsxe2x80x9d where obtaining data is difficult. They can read PSI and DELTA of a Material System over a full Range of Degrees with the only limitation being that if PSI becomes essentially zero (0.0), one can""t then determine DELTA as there is not sufficient PSI Polar Vector Length to form the angle between the PSI Vector and an xe2x80x9cXxe2x80x9d axis. In comparison, Rotating Analyzer and Rotating Polarizer Ellipsometers have xe2x80x9cDead Spotsxe2x80x9d at DELTA""s near 0.0 or 180 Degrees and Modulation Element Ellipsometers also have xe2x80x9cDead Spotsxe2x80x9d at PSI near 45 Degrees). The utility of Rotating Compensator Ellipsometer Systems should then be apparent. Another benefit provided by fixed Polarizer (P) and Analyzer (A) positions is that polarization state sensitivity to input and output optics during data acquisition is essentially non-existent. This enables relatively easy use of optic fibers, mirrors, lenses etc. for input/output.
A Search of Patents relevant to the present invention has identified very little. Most important, is a Patent to Johs et al, U.S. Pat. No. 5,872,630, from which the present invention is derived. Said Patent describes:
A spectroscopic rotating compensator material system investigation system comprising a source of a polychromatic beam of electromagnetic radiation, a polarizer, a stage for supporting a material system, an analyzer, a dispersive optics and at least one detector system which contains a multiplicity of detector elements, said spectroscopic rotating compensator material system investigation system further comprising at least one compensator(s) positioned at a location selected from the group consisting of:
before said stage for supporting a material system;
after said stage for supporting a material system; and
both before and after said stage for supporting a material system;
such that when said spectroscopic rotating compensator material system investigation system is used to investigate a material system present on said stage for supporting a material system, said analyzer and polarizer are maintained essentially fixed in position and at least one of said at least one compensator(s) is caused to continuously rotate while a polychromatic beam of electromagnetic radiation produced by said source of a polychromatic beam of electromagnetic radiation is caused to pass through said polarizer and said compensator(s), said polychromatic beam of electromagnetic radiation being also caused to interact with said material system, pass through said analyzer and interact with said dispersive optics such that a multiplicity of essentially single wavelengths are caused to simultaneously enter a corresponding multiplicity of detector elements in said at least one detector system.
Said 630 Patent also, amongst other disclosure, describes a Mathematical Regression based Calibration procedure which makes possible the use of essentially any compensator regardless of non-achromatic characteristics.
Another Patent to Johs, from which the 630 Patent was Continued-in Part, is U.S. Pat. No. 5,666,201, filed Sep. 20, 1995. The focus in said 201 Patent comprises a detector arrangement in which multiple orders of a dispersed beam of electromagnetic radiation are intercepted by multiple detector systems. However, claim 8 in the 201 Patent, in combination with a viewing the Drawings therein, provide conception of the Spectroscopic Rotating Compensator Ellipsometer, as Claimed in claim 1 of the JAW 630 Patent and, in fact, the the 630 Patent issued in view of a Terminal Disclaimer based upon the 201 Patent.
Also disclosed is U.S. Pat. No. 5,706,212, Issued Jan. 6, 1998, and Filed Mar. 20, 1996 for an Infrared Ellipsometer System Regression based Calibration Procedure. Said 212 Patent describes use of an Achromatic Rotating Compensator and application of Mathematical Regression in a Calibration procedure which evaluates calibration parameters in both rotating and stationary components. Said 630 Patent was Continued-in-Part therefrom.
A recent Patent to Aspnes, is U.S. Pat. No. 5,877,859. This Patent describes a Broadband Spectroscopic Rotating Compensator Ellipsometer System wherein the Utility is derived from selecting a Wavelength Range and Compensator so that at least one wavelength in said wavelength Range has a retardation imposed of between 135 and 225 Degrees, and another wavelength in the wavelength Range has a retardation imposed which is outside that retardation Range. The entire Utility of the Therma-wave Patent derives from that condition being met so that coefficients of two-omega and four-omega terms at various wavelengths provide information, even when other such coefficients do not.
Another Patent, U.S. Pat. No. 4,053,232 to Dill et al. describes a Rotating-Compensator Ellipsometer System, which operates utilizes monochromatic light. Two Patents which identify systems which utilize Polychromatic light in investigation of material systems are described in U.S. Pat. Nos. 5,596,406 and 4,668,086, to Rosencwaig et al. and Redner, respectively, were also identified. Also identified is a Patent to Woollam et al, U.S. Pat. No. 5,373,359 as it describes a Rotating Analyzer Ellipsometer System which utilizes white light. Patents continued from the 359 Woollam et al. Patent are, U.S. Pat. Nos. 5,504,582 to Johs et al. and U.S. Pat. No. 5,521,706 to Green et al. Said 582 Johs et al. and 706 Green et al. Patents describe use of polychromatic light in a Rotating Analyzer Ellipsometer System. A Patent to Bernoux et al., U.S. Pat. No. 5,329,357 is identified as it describes the use of optical fibers as input and output means in an ellipsometer system. A Patent to Chen et al., U.S. Pat. No. 5,581,350 is identified as it describes the application of regression in calibration of ellipsometer systems.
An article by Johs, titled xe2x80x9cRegression Calibration Method For Rotating Element Ellipsometersxe2x80x9d, which appeared in Thin Film Solids, Vol. 234 in 1993 is also identified as it predates the Chen et al. Patent and describes an essentially similar approach to ellipsometer calibration. An article by Jellison Jr. titled xe2x80x9cData Analysis for Spectroscopic Ellipsometryxe2x80x9d, Thin Film Solids, 234, (1993) is identified as it describes a method for determining the accuracy with which certain data points can be measured, which information allows adding a weighting factor to a curve fitting regression procedure as applied to a multiplicity of data points, said weighting factor serving to emphasize the effect of more accurate and precise data. A book by Azzam and Bashara titled xe2x80x9cEllipsometry and Polarized lightxe2x80x9d North-Holland, 1977 is disclosed and incorporated herein by reference for general theory. An article by Collins titled xe2x80x9cAutomated Rotating Element Ellipsometers: Calibration, Operation, and Real-Time Applicationsxe2x80x9d, Rev. Sci. Instrum. 61(8), August 1990 is identified as it provides insight into rotating element ellipsometers. An article by Kleim et al. titled xe2x80x9cSystematic Errors in Rotating-Compensator Ellipsometryxe2x80x9d published in J. Opt. Soc. Am./Vol. 11, No. 9, September 1994 is identified as it describes calibration of rotating compensator ellipsometers. An Article by An and Collins titled xe2x80x9cWaveform Analysis With Optical Multichannel Detectors: Applications for Rapid-Scan Spectroscopic Ellipsometerxe2x80x9d, Rev. Sci. Instrum., 62 (8), August 1991 is also identified as it discusses effects such as Detection System Error Characterization, Stray Light, Image Persistence etc., and calibration thereof. Also disclosed are articles by Schubert et al. which describe xe2x80x9cGeneralized Ellipsometryxe2x80x9d. The first thereof is titled xe2x80x9cExtension Of Rotating-Analyzer Ellipsometry To Generalized Ellipsometry: Determination Of The Dielectric Function Tensor From Uniaxial TiO2xe2x80x9d, J. Opt. Soc. Am. A. 13, (1996). The second such article is authored by M. Schubert alone and is titled xe2x80x9cPolarization Dependent Parameters Of Arbitrary Anisotropic Homogeneous Epitaxial Systemsxe2x80x9d, Phys. Rev. B 53, (1996). The third such article is titled xe2x80x9cGeneralized Transmission Ellipsometry For Twisted Biaxial Dielectric Media: Application To Chiral Liquid Crystalsxe2x80x9d, J. Opt. Soc. Am. A/Vol. 13, No. 9 (1996). Further identified as authority for Matrix Mathematics is a paper by Jones titled xe2x80x9cA New Calculus For The Treatment Of Optical Systemsxe2x80x9d, J.O.S.O., Vol. 31, (July 1941), and identified as describing application of Mueller Matricies in Rotating Compensator Ellispometers which utilize imperfect compensators, is a paper by Hauge titled xe2x80x9cMueller Matrix Ellipsometry With Imperfect Compensatorsxe2x80x9d, J. Opt. Soc. Am., Vol. 68, No. 11, (November 1978). As well, identified for authority regarding regression, is a book titled Numerical Recipes in xe2x80x9cCxe2x80x9d, 1988, Cambridge University Press.
Additionally, Patents pertaining to optical element retarders per se are:
U.S. Pat. No. 4,917,461 to Goldstein, which describes an achromatic infrared retarder comprised of two identical prisms in combination with a rflective surface;
U.S. Pat. No. 4,772,104 to Buhrer which describes an achromatic optical filter comprised of two birefringent disks.
U.S. Pat. No. 4,961,634 to Chipman describes an infrared achromatic retarder comprised of CdS and CdSe plates aligned with the fast axes thereof perpendicular to one another.
U.S. Pat. No. 5,946,098 to Johs, Herzinger and Green, which describes numerous optical elements.
Regarding compensators, attention is directed to the Figure in U.S. Pat. No. 548,495 to Abbe; FIG. 2 in U.S. Pat. No. 4,556,292 to Mathyssek et al.; FIGS. 1 and 4 in U.S. Pat. No. 5,475,525 Tournois et al.; and FIG. 10 in U.S. Pat. No. 5,016,980 Waldron. U.S. Pat. No. 3,817,624 to Martin and U.S. Pat. No. 2,447,828 to West were also identified.
Regarding polarizers, Patents were also identified. For instance, U.S. Pat. No. 4,221,464 to Pedinoff et al. shows a Double Plate Brewster Angle Polarizer (10) (12) which serves to avoid back reflections, which is combined with a Wire Grid Polarizer (28) present on Plate (14). A purpose of the invention is to increase the extinction coefficient of the Wire Grid Polarizer, by combining it with the Double Plate Brewster Angle Polarizer (10) (12).
A Patent, U.S. Pat. No. 5,177,635 to Keilmann, shows an Infrared Polarizer Structure of patterned metal strips on a transparent material, but does not suggest a dual polarizer arrangement.
A Patent, Patent No. 452 to Mertz, shows multiple Wire Grids oriented at angles with respect to one another. The system is an Interferometer in which said one said Wire Grid is rotated with respect to another.
A Patent, U.S. Pat. No. 3,439,968 to Hansen, shows an Infrared Brewster Angle Polarizer.
A Patent, U.S. Pat. No. 3,428,388 to Kuebler et al., shows a UV xe2x80x9cBiotitexe2x80x9d based Brewster Angle Polarizer.
A Patent, U.S. Pat. No. 5,187,611 to White et al., shows a system which illuminates an object while avoiding reflection and glare.
U.S. Pat. No. 4,733,926 to Title; U.S. Pat. No. 5,548,427 to May and U.S. Pat. No. 5,402,260 to Tsuneda et al. were also identified.
Allowed patent application Ser. No. 09/162,217 filed Sep. 29, 1998 is also identified as it teaches a method for compensating the effects of birefringent windows in an ellipsometer system.
Even in view of the foregoing, a need remains for improved Spectroscopic Rotating Compensator Material System Investigation Systems, including a Photo Array, for simultaneously detecting a Multiplicity of Wavelengths, and which can be realized utilizing off-the-shelf, non-ideal, compensators and diode array spectrometers. The present invention therefore provides a Spectroscopic Rotating Compensator Material System Investigation System which preferrably comprises a non-ideal Compensator having a preferred fast axes offset, dual zero-order waveplate compensator, and discloses alternative use of D.C. and A.C data normalization in various calibration steps.
It is generally considered that while Rotating Compensator Material System Investigation Systems, (eg. Rotating Compensator Ellipsometers), provide many benefits, (eg. Material System PSI and DELTA investigation limiting xe2x80x9cdead-spotsxe2x80x9d are not present), that in the absence of essentially Achromatic xe2x80x9cidealxe2x80x9d Compensators it would be prohibitively difficult and expensive to build, calibrate and utilize a xe2x80x9cSpectroscopicxe2x80x9d Rotating Compensator Material System Investigating System. This is to be understood in light of the fact that Compensators which are essentially Achromatic, (ie. provide essentially constant retardation over a large range of Wavelengths, such as from, (less than or equal to), 190 to 1000 or higher nanometers), are not generally and economically available as off-the-shelf items, (this being particulalry true where a Compensator is rotated during use).
The present invention system is, however, an affordable, easy to calibrate and utilize Spectroscopic Rotating Compensator Material System Investigation System comprising a Source of a Polychromatic Beam of Electromagnetic Radiation, a Polarizer, a Stage for Supporting a Material System, an Analyzer, a Dispersive Optics and at least one Photo Array Detector Element System which contains a multiplicity of Detector Elements, which Spectroscopic Rotating Compensator Material System Investigation System further comprises at least one Compensator(s) positioned at a location selected from the group consisting of: (before said stage for supporting a sample system and after said stage for supporting a sample system and both before and after said stage for supporting a sample system).
Of particular importance is the fact that said at least one Compensator(s) utilized in the present invention can, technically, be essentially any available, reasonably priced, off-the-shelf Retardation providing system, including non-Achromatic Berek-type, Zero-Order Waveplate, Multiple-Order Waveplate, Zero-Order Waveplate constructed from Multiple Multiple-Order Waveplates, Sequential Systems of Multiple Zero-Order Waveplates, each of which can be constructed from Multiple Multiple-Order Waveplates, Polymer Retarder, Mica Waveplate, Freshnel Rhomb, Achromatic, and Pseudo-Achromatic, etc. For general information, it is noted that a Berek-type Compensator is a uniaxially anisotropic plate of material in which the Optical Axis is oriented perpendicularly to a plate surface thereof. When a Polarized Beam of Electromagnetic Radiation is caused to be incident other than along the Optical Axis, orthogonal components thereof encounter different effective Indicies of Refraction, thereby effecting retardation therebetween. Polymer Compensators are made of a polymer material and can provide true Zero-Order retardance which, as do many Compensators, provides an inverse wavelength functional Retardance Characteristic. Essentially Achromatic (Pseudo-Achromatic) Compensators can be constructed by stacking appropriately chosen Polymer and Crystal waveplates. A potential advantage of said essentially Achromatic Compensators is that Retardance can be essentially constant over a range of wavelengths.
Sequential Systems of Multiple Zero-Order Waveplates allow achieving flattened Retardance vs. Wavelength characteristics, and, it is noted, are the preferred present invention Compensator type. To ellaborate, the preferred present invention Compensator system comprises a system of at least two (ie. First and Second), Zero-Order Waveplates, each of which Zero-Order Waveplates can be a single plate, (eg. mica or polymer), or constructed from an effective combination of Multiple-Order Waveplates, (eg. two quartz plates or bicrystaline waveplates such as Cadnium Sulfide or Cadnium Selenide). As further insight, an effective Zero-Order Waveplate can be functionally constructed by combining two Multi-Order (eg. Quartz) Waveplates which have Optical Axes oriented at a nominal ninety (90) degrees with respect to one another. That is, two Multi-Order waveplates are selected and combined so that the difference in retardation entered by each gives rise to an overall Zero-Order Waveplate retardance characteristic. In particular, the prefered present invention Compensator embodiment provides that each of said First and Second effectively Zero Order Waveplates be formed by physically optically combining two Multiple Order Waveplates, such that the net result of passing a beam of electromagnetic radiation therethrough is essentially equivalent to the result which would achieved by passing said electromagnetic beam through a single plate Zero-Order Waveplate. The reason that such effective Zero-Order Waveplates, which are formed by physically combining two Multiple Order Waveplates are preferred, is that such effectively Zero-Order Waveplates are readily and economically available in the marketplace, and that true single plate Zero-Order Waveplates are typically physically delicate and difficult to utilize. Continuing, the preferred present invention Compensator provides that two of said per se. or effectively Zero-Order Waveplate Compensators be oriented with respect to one another such that the fast axes of the First per se. or effectively Zero-Order Compensator are rotated with respect to the Second per se. or effectively Zero-Order Compensator, away from zero or ninety degrees, and typically within some range around a nominal forty-five (45) degrees. In use, a beam of electromagnetic radiation utilized to investigate a material system, is caused to pass through both of said First and Second Compensators with the result achieved being that a present invention preferred Compensator configuration provides a quasi-achromatic retardation of between, for instance, forty (40) and one-hundred-thirty (130) degrees over relatively large wavelength ranges within one-hundred-ninety (190 NM) to seventeen-hundred (1700 NM). That is, preferred present invention Compensators are specifically designed to provide retardation values which never exceed one-hundred-eighty (180) degrees, or even one-hundred-thirty-five (135) degrees at any utilized wavelength. It is noted that this is in direct contrast to the practice of Therma-wave rotating compensator systems as described in U.S. Pat. No. 5,877,859 to Aspnes, wherein large chromaticities in compensator systems utilized are preferrable so that various two-omega and four-omega term coefficients are available at various wavelengths, emphasis added.
The present invention then, preferably utilizes at least one of compensator selected from the group consisting of:
being comprised of at least two zero-order waveplates, said zero-order waveplates having their respective fast axes rotated to a position offset from zero or ninety degrees with respect to one another;
being comprised of a combination of at least a first and a second effective zero-order wave plate, said first effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another, and said second effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another; the fast axes of the multiple order waveplates in said second effective zero-order wave plate being rotated to a position at a nominal forty-five degrees to the fast axes of the multiple order waveplates in said first effective zero-order waveplate;
being comprised of a combination of at least a first and a second effective zero-order wave plate, said first effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another, and said second effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another; the fast axes of the multiple order waveplates in said second effective zero-order wave plate being rotated to a position away from zero or ninety degrees with respect to the fast axes of the multiple order waveplates and in said first effective zero-order waveplate;
comprised of at least one zero-order waveplate and one effective zero-order waveplate, said effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another, the fast axes of the multiple order waveplates in said effective zero-order wave plate being rotated to a position away from zero or ninety degrees with respect to the fast axis of the zero-order waveplate.
(Note, zero-order and effective zero-order waveplates are of, for instance, single plate and multiple waveplate construction respectively).
Continuing, because the preferred present invention Compensators do not provide an exact Ninety (90) Degrees of Retardation at all wavelengths over a relatively large range of Wavelengths, the present invention, as described supra herein, utilizes a Regression based Calibration procedure which compensates for said non-ideal Compensator Retardation characteristics. And while it is true that the sensitivity and accuracy of a Rotating Compensator Material System Investigation System degrades as the Retardance provided by a utilized Compensator approaches zero (0.0) or one-hundred-eighty (180) degrees, again, it has been found that Compensators which demonstrate Retardation over a range of utilized Wavelengths, of from forty (40) to less than one-hundred-thirty-five (135) degrees, (thereby avoiding infringement of recent U.S. Pat. No. 5,877,859 to Aspnes et al.), are readily available, or can be constructed from readily available components, which are very acceptable for use in the present invention Rotating Compensator Ellipsometer System, and said Compensators enable achieving very impressive results over a demonstrated relatively large range of wavelengths, (eg. at least two-hundred-fifty (250) to one-thousand (1000) nanometers). A present invention spectroscopic rotating compensator material system investigation system typically comprises at least one compensator(s) which produces a retardance of, preferably, between seventy-five (75) and one-hundred-thirty (130) degrees over a range of wavelengths defined by a selection from the group consisting of:
a. between one-hundred-ninety (190) and seven-hundred-fifty (750) nanometers;
b. between two-hundred-forty-five (245) and nine-hundred (900) nanometers;
c. between three-hundred-eighty (380) and seventeen-hundred (1700) nanometers;
d. within a range of wavelengths defined by a maximum wavelength (MAXW) and a minimum wavelength (MINW) wherein the ratio of (MAXW)/(MINW) is at least one-and-eight-tenths (1.8).
Acceptable practice however, provides for the case wherein at least one of said at least one compensator(s) provides a retardation vs. wavelength characteristic retardation between thirty (30.0) and less than one-hundred-thirty-five (135) degrees over a range of wavelengths specified from MINW to MAXW by a selection from the group consisting of:
a. MINW less than/equal to one-hundred-ninety (190) and MAXW greater than/equal to seventeen-hundred (1700) nanometers;
b. MINW less than/equal to two-hundred-twenty (220) and MAXW greater than/equal to one-thousand (1000) nanometers;
c. within a range of wavelengths defined by a maximum wavelength (MAXW) and a minimum wavelength (MINW) range where (MAXW)/(MINW) is at least four-and-one-half (4.5).
(NOTE, the specified vales and ranges can not be achieved by single plates with (1/wavelength) retardation characteristics).
Continuing, when the present invention Spectroscopic Rotating Compensator Material System Investigation System is used to investigate a Material System present on said Stage for Supporting a Material System, said Analyzer and Polarizer are maintained essentially fixed in position and at least one of said at least one Compensator(s) is/are caused to continuously rotate while a Polychromatic Beam of Electromagnetic Radiation produced by said Source of a Polychromatic Beam of Electromagnetic Radiation is caused to pass through said Polarizer and said Compensator(s). Said Polychromatic Beam of Electromagnetic Radiation is also caused to interact with said Material System, pass through said Analyzer and interact with said Dispersive Optics such that a Multiplicity of Essentially Single Wavelengths are caused to simultaneously enter a corresponding multiplicity of Detector Elements in said Detector System Photo Array.
A method of calibrating a present invention Spectroscopic Rotating Compensator Material System Investigation System can comprise the steps of:
a. providing a present invention Spectroscopic Rotating Compensator Material System Investigation System as just described infra herein.
b. developing a Mathematical Model of said Spectroscopic Rotating Compensator Material System Investigation System which comprises as Calibration Parameter variables Polarizer Azimuthal Angle Orientation, present Material System PSI""s and present Material System DELTA""s vs. wavelength, Compensator Azimuthal Angle Orientation(s), Matrix Components of said Compensator(s), Analyzer Azimuthal Angle Orientation, and optionally Detector Element Image Persistance and Readout non-Idealities, which Mathematical Model is effectively a Transfer Function which enables calculation of Electromagnetic Beam Intensity as a function of Wavelength detected by a Detector Element, given Intensity as a function of wavelength provided by said Source of a Polychromatic Beam of Electromagnetic Radiation, said Mathematical Model optionally providing equations for Coefficients of Terms in said Transfer Function, said Coefficients of terms being functions of Calibration Parameters;
c. causing a Polychromatic Beam of Electromagnetic Radiation produced by said Source of a Polychromatic Beam of Electromagnetic Radiation, to pass through said Polarizer, interact with a Material System caused to be in the path thereof, pass through said Analyzer, and interact with said Dispersive Optics such that a Multiplicity of Essentially Single Wavelengths are caused to simultaneously enter a corresponding Multiplicity of Detector Elements in said at least one Detector System, with said Polychromatic Beam of Electromagnetic Radiation also being caused to pass through said Compensator(s) positioned at a location selected from the group consisting of: (before said Stage for Supporting a Material System and after said Stage for Supporting a Material system and both before and after said Stage for Supporting a Sample System);
d. obtaining an at least Two Dimensional Data Set of Intensity Values vs. Wavelength and a parameter selected from the group consisting of: (Angle-Of-Incidence of said Polychromatic Beam of Electromagnetic Radiation with respect to a present Material System, and Azimuthal Angle Rotation of one element selected from the group consisting of: (said Polarizer and said Analyzer)), over time, while at least one of said at least one Compensator(s) is caused to continuously rotate and, optionally, from said data set calculating numerical values for Coefficients of Terms in the Transfer Function for said Spectroscopic Rotating Compensator Material System Investigation System;
e. performing a Mathematical Regression of said Mathematical Model onto said at least Two Dimensional Data Set and/or onto values for Coefficients of Terms in the Transfer Function to evaluate said Calibration Parameters;
said Regression based Calibration Procedure evaluated Calibration Parameters serving to compensate said Mathematical Model for non-Achromatic characteristics and non-Idealities of said Compensator(s), and for Azimuthal Angle Orientations of said Polarizer, Analyzer and Compensator(s).
In addition, evaluation of Detector System Detector Element Image Persistance and Readout non-Ideality compensation Calibration Parameters also included in the Mathematical Model, can simultaneously be carried out in the Mathematical Regression Procedure.
It is noted that where two Compensators are present, each can be rotated at essentially the same, or different speeds.
Said Method of Calibrating a Spectroscopic Rotating Compensator Material System Investigation System can include, in the step of providing the Dispersive Optics and Photo Array Detector Elements of the Spectroscopic Rotating Compensator Material System Investigation System, the step of effecting common mounting thereof with a Beam Splitting Means, said Beam Splitting Means serving to divert a portion of the Polychromatic Beam of Electromagnetic Radiation which otherwise proceeds to said Dispersive Optics, and transmit the remainder of said Polychromatic Beam of Electromagnetic Radiation toward said Dispersive Optics. In use, the diverted portion of said Polychromatic Beam of Electromagnetic Radiation can then be directed by said Beam Splitting Means into an Alignment Means, (which can be a Cross-Hairs or an Automated Polychromatic Beam of Electromagnetic Radiation Detecting and commonly mounted Dispersive Optics and Detector Elements Orientation Control Means). In use said Alignment Means then provides monitored Alignment Capability thereby allowing precise control of the Locus of Propagation of the portion of said Polychromatic Beam of Electromagnetic Radiation which passes through said Beam Splitting Means, interacts with said Dispersive Optics, and enters said Detector Elements.
Said Method of Calibrating a Spectroscopic Rotating Compensator Material System Investigation System can also include, in the step of providing a Mathematical Model, the steps of providing a Matrix Representation of each of said Polarizer, present Material System, said Compensator(s), and said Analyzer, and determining a Mathematical Transfer Function relating Electromagnetic Beam Intensity Out to Intensity In, as a function of Wavelength, by multiplication of said Matrices in a Spectroscopic Rotating Compensator Material System Investigation System element presence representing order.
In what directly follows, a recitation of practice taught in U.S. Pat. No. 5,872,630 is reviewed. Following thereafter is disclosure of revised present invention practice.
Said Method of Calibrating a Spectroscopic Rotating Compensator Material System Investigation System can involve, in the step of calculating values of Coefficients of a Transfer Function from said Data Set, the calculation of values of Coefficients of a Fourier Series, (eg.xcex12, xcex22, xcex14, xcex24, in Eqs. 11-14 supra).
Additionally, said Method of Calibrating a Spectroscopic Rotating Compensator Material System Investigation system can further comprise the step of Parameterizing Calibration Parameters by representing variation as a function of Wavelength, (or perhaps Angle-Of-Incidence of said Polychromatic Beam of Electromagnetic Radiation with respect to a Surface of an Investigated Material System or Other Variable), by a Calibration Parameter containing Mathematical Equation, Calibration Parameter(s) in said Calibration Parameter containing Mathematical Equation being evaluated during said Mathematical Regression. (See Eqs. 50 and 51 below). When this is done the Calibration Parameter containing Mathematical Equation provides a functional relationship, and, it is noted, can even be a constant value over a range of, for instance, Wavelengths and/or Polarizer Azimuthal Angle settings). (Note, said parameterized approach to mathematical regression based calibration parameter evaluation is better described supra herein under the Headings GLOBAL REGRESSION MODES 1, 2 and 3).
It is further noted that the at least Two Dimensional Data Set can be obtained with the Spectroscopic Rotating Compensator Material System Investigation System oriented in a xe2x80x9cStraight-Throughxe2x80x9d or xe2x80x9cMaterial-System-Presentxe2x80x9d configuration. In the first configuration open atmosphere essentially constitutes a material system, and a Polarized Electromagnetic Beam passes directly through the Polarizer, Compensator(s) and Analyzer into the Detector System. In the second configuration a Material System is present which presents PSI and DELTA values other than those of the open atmosphere so that a Polychromatic Electromagnetic Beam passes through the Polarizer, possibly a Compensator, and then interacts with a Material System, before passing through, possibly a Compensator, an Analyzer and into the Detector System. Compensator(s), it should be understood, can be present before and/or after the Material System.
With the above general description of the present invention System and Calibration Method in mind, attention is directed to providing a detailed demonstration of the Calibration Procedure of the present invention as applied to a Spectroscopic Rotating Compensator Ellipsometer System sequentially comprised of:
A Polychromatic Light Source
A Fixed Polarizer
A Material Sample
A Continuously Rotating Compensator
A Fixed Analyzer, and
A Detector Element containing Photo Array.
(Note: the Reflection mode side of FIG. 1 of this Disclosure shows this basic configuration where Compensator (C) is considered as removed and only Compensator (Cxe2x80x2) remains present).
It is to be appreciated, however, that the basic approach to calibration described directly, is adaptable for use in systems in which the Continuously Rotating Compensator is placed ahead of a Material System, and in systems in which two Compensators are present, one ahead of, and one after a Material System wherein one or both are caused to Continuously Rotate in use. For instance, in the case where a Rotating Compensator is placed ahead of the Material Sample, rather than thereafter, simply exchanging references to Polarizer and Analyzer in equations derived for the case where the Rotating Compensator is placed after the Material System, provides the applicable equations.
Transfer function equations for the Rotating Compensator system configured as recited above can be obtained from multiplication of Matrix Representations of the various components, in an appropriate order, in conjunction with Trig function containing Rotation Matrices, which serve to align coordinate systems between components. Eq. 1 shows said Matrix representation:                               E          ⁡                      (                          P              ,              ψ              ,              Δ              ,              C              ,              r1              ,              r2              ,              r3              ,              r4              ,              A                        )                          =                              (                                                            1                                                  0                                                                              0                                                  0                                                      )                    ·                      (                                                                                cos                    ⁡                                          (                      A                      )                                                                                                            sin                    ⁡                                          (                      A                      )                                                                                                                                        -                                          sin                      ⁡                                              (                        A                        )                                                                                                                                  cos                    ⁡                                          (                      A                      )                                                                                            )                    ·                      (                                                                                cos                    ⁡                                          (                      A                      )                                                                                                            sin                    ⁡                                          (                      A                      )                                                                                                                                        -                                          sin                      ⁡                                              (                        A                        )                                                                                                                                  cos                    ⁡                                          (                      A                      )                                                                                            )                    ·                      (                                                            r1                                                  r3                                                                              r2                                                  r4                                                      )                    ·                      (                                                                                cos                    ⁡                                          (                      C                      )                                                                                                            sin                    ⁡                                          (                      C                      )                                                                                                                                        -                                          sin                      ⁡                                              (                        C                        )                                                                                                                                  cos                    ⁡                                          (                      C                      )                                                                                            )                    ·                      (                                                                                sin                    ⁢                                          xe2x80x83                                        ⁢                                          Ψ                      ·                                              ⅇ                                                  1                          ⁢                          k                          ⁢                                                      xe2x80x83                                                    ⁢                          Δ                                                                                                                                      0                                                                              0                                                                      cos                    ⁢                                          xe2x80x83                                        ⁢                    Ψ                                                                        )                    ·                      (                                                                                cos                    ⁡                                          (                      P                      )                                                                                                                                        sin                    ⁡                                          (                      P                      )                                                                                            )                                              (        1        )            
where: xcexa8 and xcex94 are the traditional ellipsometric parameters which describe the Material System;
P is the azimuthal orientation of the Polarizer;
C is the azimuthal orientation of the Rotating Compensator;
r1, r2, r3 and r4 are the Jones Matrix elements which describe the Compensator, (Note that a Jones Matrix is utilized, however, a Mueller Matrix or other Matrix could also be utilized);
A is the azimuthal orientation of the Analyzer.
The Light Intensity which is measured by a Detector is provided by multiplying through the Matrices in Eq. 1 to provide a Complex Result, then multiplying said Complex Result by its Complex Conjugate. Eq. 2 indicates this:
I(P,xcexa8,xcex94, C,r1,r2,r3,r4,A)=E(P,xcexa8,xcex94,C,r1,r2,r3,r4,A)xc2x7E*(P,xcexa8,xcex94,C,r1,r2,r3,r4,A)xe2x80x83xe2x80x83(2)
The Intensity Equation I(t), (Eq. 8):
I(t)=I0(DC+xcex12 cos 2C+xcex22 sin 2C+xcex14 cos 4C+xcex24 sin 4C)xe2x80x83xe2x80x83(8)
which results from said multiplication is very involved, but can be expressed in terms of intermediate results as provided in Eq. 3-7, via Eqs. 9.
p1=sin xcexa8xc2x7(cos xcex94+ixc2x7sin xcex94)xc2x7cos P
p2=cos xcexa8xc2x7sin Pxe2x80x83xe2x80x83(3)
K1=(xe2x88x92p1xc2x7r3+p2xc2x7r1)
K2=(p1xc2x7r1+p2xc2x7r3)
K3=(xe2x88x92p1xc2x7r4+p2xc2x7r2)
K4=(p1xc2x7r2+p2xc2x7r4)xe2x80x83xe2x80x83(4)
xe2x80x83U1=(cos(A)xc2x7K2+sin(A)xc2x7K4)
U2=(K3+K2)xc2x7sin(A)+(K1xe2x88x92K4)xc2x7cos(A)
U3=(cos(A)xc2x7K3+sin(A)xc2x7K1)xe2x80x83xe2x80x83(5)
                              V1          =                                                    U1                ·                                  U1                  _                                            ⁢                              xe2x80x83                            ⁢              V2                        =                                                            U2                  ·                                      U2                    _                                                  ⁢                                  xe2x80x83                                ⁢                V3                            =                              U3                ·                                  U3                  _                                                                    ⁢                  
                ⁢                  V4          =                                                    2                ·                                  Re                  ⁡                                      (                                          U1                      ·                                              U2                        _                                                              )                                                              ⁢                              xe2x80x83                            ⁢              V5                        =                          2              ·                              Re                ⁡                                  (                                      U1                    ·                                          U3                      _                                                        )                                                                    ⁢                  
                ⁢                  V6          =                      2            ·                          Re              ⁡                              (                                  U2                  ·                                      U3                    _                                                  )                                                                        (        6        )                                          T1          =                                    V1              +                              V3                ⁢                                  xe2x80x83                                ⁢                T2                                      =                                          V2                +                                  V5                  ⁢                                      xe2x80x83                                    ⁢                  T3                                            =                              V1                -                V3                                                    ⁢                  
                ⁢                  T4          =                                    V4              +                              V6                ⁢                                  xe2x80x83                                ⁢                T5                                      =                          V4              -              V6                                                          (        7        )            
where Eqs. 9 provide that:                                           D            ⁢                          xe2x80x83                        ⁢            C                    =                                                    3                8                            ·              T1                        +                                          1                8                            ·              T2                                      ⁢                  
                ⁢                              α            2                    =                                                                      1                  2                                ·                T3                            ⁢                              xe2x80x83                            ⁢                              β                2                                      =                                          1                4                            ·              T4                                      ⁢                  
                ⁢                              α            4                    =                                                                      1                  8                                ·                                  (                                      T1                    ⁢                                          xe2x80x83                                        ⁢                    T2                                    )                                            ⁢                              xe2x80x83                            ⁢                              β                4                                      =                                          1                8                            ·              T5                                                          (        9        )            
and C=xcfx89xc2x7t, where xe2x80x98xcfx89xe2x80x99 is the angular frequency of the continuously rotating Compensator and I0 is an arbitrary constant.
(It is further noted that Eq. 8 is a truncated Fourier Series, and could include additional, higher harmonic terms).
Equations 1-9 are appropriate for a Material System which does not depolarize an Electromagnetic Beam used to investigate a Material System, such that Jones Matrix formalism is appropriate. If a Material System is investigated which does depolarize an investigation electromagnetic beam, then Mueller Matrix formalism can be substituted. As well, the xe2x80x9cIsotropicxe2x80x9d Material System Matrix in Eq. 1 could be replaced by a General Material System Matrix in the Rotating Compensator. This is described by M. Schubert in the context of xe2x80x9cGeneralized Ellipsometryxe2x80x9d, (see Background Section for citations to relevant articles which treat the topic of Generalized ellipsometry by Schubert).
If an ideal Compensator is assumed, where the Jones Matrix components are:
r1=1;
r2=0;
r3=0; and
r4=eixe2x88x92xcex4;
then the Eqs. 9 become Eqs 10-14:
DC=(xc2xd)(1+cos xcex4)[cos 2A(cos 2Pxe2x88x92cos 2xcexa8)+sin 2A sin 2P sin 2xcexa8 cos xcex94]xe2x88x92cos 2P cos 2xcexa8+1xe2x80x83xe2x80x83(10)
xcex12=xe2x88x92sin 2A sin 2P sin xcex4 sin 2xcexa8 sin xcex94xe2x80x83xe2x80x83(11)
xcex22=cos 2A sin 2P sin xcex4 sin 2xcexa8 sin xcex94xe2x80x83xe2x80x83(12)
604=(xc2xd)(1xe2x88x92cos xcex4) [cos 2A(cos 2Pxe2x88x92cos 2xcexa8)xe2x88x92sin 2A sin 2P sin 2xcexa8 cos xcex94]xe2x80x83xe2x80x83(13)
xcex24=(xc2xd) (1xe2x88x92cos xcex4) [sin 2A(cos 2Pxe2x88x92cos 2xcexa8)+cos 2A sin 2P sin 2xcexa8 cos xcex94]xe2x80x83xe2x80x83(14)
It is noted that said Eqs. 10-14 are found in Kleim et al. as referenced in the Background Section of this Specification, with xe2x80x9cAxe2x80x9d and xe2x80x9cPxe2x80x9d interchanged. (The Kleim et al. work assumed a Rotating Compensator present prior to a Material System).
Continuing, Eqs. 10-14 are valid for an ideal Rotating Compensator System wherein the Azimuthal angles of the optics are perfectly aligned with the Material Sample frame of reference. In practice this is never true, and offset terms xe2x80x9cAxe2x80x2xe2x80x9d, xe2x80x9cPxe2x80x2xe2x80x9d and xe2x80x9cCxe2x80x2xe2x80x9d must be entered to provide Eqs. 15a and 15b:
A=Axe2x80x2xe2x88x92As, P=Pxe2x80x2xe2x88x92Psxe2x80x83xe2x80x83(15a)
C=Cxe2x80x2xe2x88x92Csxe2x80x83xe2x80x83(15b)
where the Axe2x80x2, Cxe2x80x2 and Pxe2x80x2 indicate dial readings and the As, Cs and Ps indicate Offset Angles to be determined by a Calibration Procedure.
Substituting Eq. 15b into Eq. 8 provides Eqs. 16a and 16b, and 17a and 17b for Fourier Coefficients, (note that the DC term is unchanged):
mxcex12=xcex12 cos 2Csxe2x88x92xcex22 sin 2Csxe2x80x83xe2x80x83(16a)
mxcex22=xcex12 sin 2Cs+xcex22 cos 2Csxe2x80x83xe2x80x83(16b)
mxcex14=xcex14 cos 4Csxe2x88x92xcex24 sin 4Csxe2x80x83xe2x80x83(17a)
mxcex24=xcex14 sin 4Cs+xcex24 cos 4Csxe2x80x83xe2x80x83(17b)
Continuing, the present invention simultaneously measures the Intensity, (any functionally similar magnitude to be considered equivalent for the purposes of this disclosure), vs. time or compensator rotation angle of a multiplicity of essentially single wavelengths with a Photo Array, to determine Fourier Coefficients. And as the Diode Elements in the Photo Array are operated in a Charge Integration Mode, it is necessary to utilize a Hadamard analysis of the signal. In the embodiment of the present invention disclosed in the Parent U.S. Pat. No. 5,872,630, the Diode Array was disclosed as synchronously read-out exactly sixteen (16) times during each rotation of the Rotating Compensator. (See supra herein wherein it is reported that presently preferred practice is to read-out a photo-array an odd number, (eg. thirteen (13) times), during each rotation of the Rotating Compensator). The time varying signal, which results from modulation imposed by the Rotating Compensator, is given by Eq. 18. Eq. 19 represents a measured value at a given channel in a Photo Array for the i""th scan measured during the rotation.
s(t)=I0xc2x7(DC+xcex12 cos 2t+xcex22 sin 2t+xcex14 cos 4t+xcex24 sin 4t)xe2x80x83xe2x80x83(18)
                              h          i                =                              ∫                                          (                                  i                  -                  1                                )                            ·                              π                8                                                    i              ·                              π                8                                              ⁢                                    s              ⁡                              (                t                )                                      ⁢                          ⅆ              t                                                          (        19        )            
Substituting Eq. 18 into Eq. 19 and rearranging terms provides the following expressions, (Eqs. 20-24), for the Fourier Coefficients:                               D          ⁢                      xe2x80x83                    ⁢          C                =                                                                                                  h                    1                                    +                                      h                    2                                    +                                      h                    3                                    +                                      h                    4                                    +                                      h                    5                                    +                                      h                    6                                    +                                      h                    7                                    +                                      h                    8                                    +                                      h                    9                                    +                                                                                                                          h                    10                                    +                                      h                    11                                    +                                      h                    12                                    +                                      h                    13                                    +                                      h                    14                                    +                                      h                    15                                    +                                      h                    16                                                                                            4            ·            π            ·                          I              0                                                          (        20        )                                          α          2                =                                                                                                  h                    1                                    +                                      h                    2                                    -                                      h                    3                                    -                                      h                    4                                    -                                      h                    5                                    -                                      h                    6                                    +                                      h                    7                                    +                                      h                    8                                    +                                      h                    9                                    +                                                                                                                          h                    10                                    -                                      h                    11                                    -                                      h                    12                                    -                                      h                    13                                    -                                      h                    14                                    +                                      h                    15                                    +                                      h                    16                                                                                            8            ·                          I              0                                                          (        21        )                                          β          2                =                                                                                                  h                    1                                    +                                      h                    2                                    +                                      h                    3                                    +                                      h                    4                                    -                                      h                    5                                    -                                      h                    6                                    -                                      h                    7                                    -                                      h                    8                                    +                                      h                    9                                    +                                                                                                                          h                    10                                    +                                      h                    11                                    +                                      h                    12                                    -                                      h                    13                                    -                                      h                    14                                    -                                      h                    15                                    -                                      h                    16                                                                                            8            ·                          I              0                                                          (        22        )            
Equations 20-24 provide means for extracting the Fourier Coefficients for the Rotating Compensator modulated signal from the (hi) values which are measured by the Photo Array Diode Elements during continuous rotation of the Rotating Compensator.
The foregoing Eqns. 18-24, and associated text, provides disclosure regarding application of Hadamard Analysis in Parent U.S. Pat. No. 5,872,630. At this point, regarding Hadamard Analysis, it is disclosed that since said disclosure was formulated, additional work has provided insight to a method for determining Hadamard Coefficients for an arbitrary xe2x80x9cnxe2x80x9d-point Hadamard Transform. It is noted that current practice is to select xe2x80x9cnxe2x80x9d to be an odd number of at least nine (9), (rather than the previously recited value of sixteen (16)), and preferably thirteen (13). This allows acounting for not only D.C. and even harmonics, (eg. second (2nd) and forth (4th) which appear in present invention output signals), but also for odd harmonics (eg. fifth (5th) and sixth (6th)), and the results of electronics non-linearities. In any case, it is emphasised that present invention practice teaches that xe2x80x9cnxe2x80x9d should be selected to be an odd number.
Continuing, present invention practice provides that that for a given sampling period, application of the Hadamard analysis leads to calculation of the D.C. Signal Intensity, plus Sin and Cos Terms at (nxe2x88x921)/2 harmonic frequencies. Consider that all measured points are indexed by a variable xe2x80x9cixe2x80x9d, while xe2x80x9cicxe2x80x9d and xe2x80x9cisxe2x80x9d index the Cos and Sin components, respectively, and xe2x80x9cjxe2x80x9d indexes harmonic frequency components.
Consider, for (n=9):
i=0 . . . nxe2x88x921;
ic=1,3 . . . nxe2x88x921
is=2,4 . . . nxe2x88x921;   j  =      0    ⁢          xe2x80x83        ⁢    …    ⁢          xe2x80x83        ⁢                            (                      n            -            1                    )                2            .      
Sin and Cos Basis Functions are piecewise integrated using Hadamard Formalism. The resulting Coefficients are then packed into a Square Matrix denoted xe2x80x9cMxe2x80x9d, as indicated in Eqs. 19:                                           hc            ⁡                          (                              i                ,                j                            )                                :=                                                    ∫                                                                            2                      ⁢                      π                                        n                                    ·                  i                                                                                                                    2                        ⁢                        π                                            n                                        ·                    i                                    +                  1                                            ⁢                                                cos                  ⁡                                      (                                          j                      ·                      t                                        )                                                  ⁢                                  ⅆ                  t                                ⁢                                  xe2x80x83                                ⁢                                  hs                  ⁡                                      (                                          i                      ,                      j                                        )                                                                        :=                                          ∫                                                                            2                      ⁢                      π                                        n                                    ·                  i                                                                                            2                      ⁢                      π                                        n                                    ·                                      (                                          i                      +                      1                                        )                                                              ⁢                                                sin                  ⁡                                      (                                          j                      ·                      t                                        )                                                  ⁢                                  ⅆ                  t                                                                    ⁢                  
                ⁢                              M                          i              ,              0                                :=                                                    hc                ⁡                                  (                                      i                    ,                    0                                    )                                            ⁢                              xe2x80x83                            ⁢                              M                                  i                  ,                  ic                                                      :=                          hc              ⁡                              (                                  i                  ,                                                            floor                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  ic                          2                                                )                                                              +                    1                                                  )                                                    ⁢                  xe2x80x83                ⁢                  
                ⁢                              M                          i              ,              is                                :=                      hs            ⁡                          (                              i                ,                                  floor                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          is                      2                                        )                                                              )                                                          19        xe2x80x2            
The xe2x80x9cMxe2x80x9d Matrix is then inverted, yielding
H:=Mxe2x88x921
a Matix of Coefficients xe2x80x9cHxe2x80x9d which can be applied to a general n-point data system stream to evaluate the xe2x80x9cnxe2x80x9d frequency components, ((D.C.+Sin+Cos terms at (nxe2x88x921)/2 frequencies).
To extract the Eq. 18 frequency components of s(t), the signal is piecewise integrated, the resulting coefficients stored in xe2x80x9chxe2x80x9d, and multiplied times the Hadamard Ceofficient Matrix xe2x80x9cHxe2x80x9d, as indicated in Eqs. labeled 20xe2x80x2-24xe2x80x2 below:
As an example consider the signal xe2x80x98s(t)xe2x80x99 below which contains a DC value of 2.5, a 2nd harmonic xe2x80x98cosxe2x80x99 term of xe2x88x926.1, and a 4th harmonic xe2x80x98sinxe2x80x99 term of 1.4.
s(t):=2.5+xe2x88x926.1xc2x7cos(2xc2x7t)+1.4xc2x7sin(4xc2x7t)
To extract the frequency components of s(t), the signal is piecewise integrated (the resultin coefficients are stored in xe2x80x98hxe2x80x99), and multiplied times the Hadamard coefficient matrix xe2x80x98Hxe2x80x99.                                                                         h                i                            :=                                                ∫                                                                                    2                        ⁢                        π                                            n                                        ·                    i                                                                                                      2                        ⁢                        π                                            n                                        ·                                          (                                              i                        +                        1                                            )                                                                      ⁢                                                      s                    ⁡                                          (                      t                      )                                                        ⁢                                      ⅆ                    t                                                                                                                                                            H                  ·                  h                                =                                                      [                                                                                            2.5                                                                                                                      0                                                                                                                      0                                                                                                                                                  -                            6.1                                                                                                                                                0                                                                                                                      0                                                                                                                      0                                                                                                                      0                                                                                                                      1.4                                                                                      ]                                    ⁢                                      xe2x80x83                                    ⁢                                                                                                              D                          ⁢                                                      xe2x80x83                                                    ⁢                          C                          ⁢                                                      xe2x80x83                                                    ⁢                          component                                                                                                                                                                                          1                            ⁢                            st                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    cos                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                                                                                    1                            ⁢                            st                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    sin                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                                                                                    2                            ⁢                            nd                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    cos                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                                                                                    2                            ⁢                            nd                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    sin                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                                                                                    3                            ⁢                            rd                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    cos                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                                                                                    3                            ⁢                            rd                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    sin                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                                                                                    4                            ⁢                            th                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    cos                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                                                                                    4                            ⁢                            th                            ⁢                                                          xe2x80x83                                                        ⁢                            harmonic                                                    ,                                                                                    sin                                                            xe2x80x2                                                                                              ‵                                                        ⁢                                                          xe2x80x83                                                        ⁢                            component                                                                                                                                                          ⁢                              xe2x80x83                                                                                  20          xe2x80x2                ⁢                  –24          xe2x80x2                    
As in the 630 Parent Patent, it is generally emphasized that good quality electronics which employ the Video Integration Read-Out technique have been found to be very conducive to accurately measuring Fourier Coefficients using Photo Array Diode Elements. It is to be understood that said good quality electronics interface output signals from Photo Array Diode Elements to a computer system which collects and analyzes data. Preferred xe2x80x9cOff-The-Shelf-Systemsxe2x80x9d which include good quality electronics, suitable for use in the present invention Rotating Compensator Material System Investigation System, are Zeiss Diode Array Spectrometer systems identified by manufacturer numbers selected from the group: MMS1 (300-1150 nm); UV/VIS MMS (190-730 nm); UV MMS (190-400 nm); AND IR MMS (900-2400 nm). Said Zeiss systems also include Dispersive Optics and Diode Element containing Photo Arrays. The Zeiss systems include fourteen (14) bit dynamic range readout electronics, which provides a voltage pulse output. The present invention system provides additional good-quality electronics in the form of an integrator and Analog to Digital Converter. In use, the scanning rate of Diode Elements in a Zeiss system Photo Array is synchronized with the rotation of the Rotating Compensator of the present invention Rotating Compensator Material System Investigation System. Said synchronization is accomplished utilizing standard digital logic, and Diode Elements in the Photo Array are scanned sixteen (16) times under previous practice, and thirteen (13) times under present practice, during each rotation of the Rotating Compensator. It is further noted that the present invention preferably effects rotation of the Rotating Compensator with a hollow shaft Stepper Motor. A sequence of reference pulses is generated by a sensor attached to the Rotating Compensator, with said reference pulses being provided to good quality electronics simultaneous with the data provided by the Photo Array Diode Elements. Said reference pulses allow correlation of the angular position of the Rotating Compensator with data provided by the Photo Array Diode Elements.
Regarding Photo Array data, it is further noted that authors, An and Collins, describe some of the non-idealities which can be present when using a Photo Array Detector in a Spectroscopic Rotating Compensator Material System Investigation System. With the exception of the An and Collins correction for xe2x80x9cStray Lightxe2x80x9d (see An and Collins Eq. 13), however, none of the Photo Array non-ideality corrections which were presented in their paper were found necessary in implementing the preferred embodiment of the present invention. However, to allow a non-ideal Photo Array to be used in the present invention, the relevant corrections for a Image Persistence, and for Read Time in a Spectroscopic Rotating Compensator Material System Investigation System in which sixteen (16) Diode Element Scans are acquired for each Rotating Compensator revolution were derived, and are provided in Eqs. 25-34.
Image Persistence correction, where xe2x80x98xxe2x80x99 is the magnitude of the non-ideality:
ipxcex12=xcex120.5xc2x7xxc2x7[(2xe2x88x92{square root over (2)})xc2x7xcex12+{square root over (2)}xc2x7xcex22]xe2x80x83xe2x80x8325)
xe2x80x83ipxcex22=xcex22xe2x88x920.5xc2x7xxc2x7[(2xe2x88x92{square root over (2)})xc2x7xcex22xe2x88x92{square root over (2)}xc2x7xcex12]xe2x80x83xe2x80x83(26)
ipxcex14=xcex14xe2x88x92xxc2x7(xcex14+xcex24)xe2x80x83xe2x80x83(27)
ipxcex24=xcex24xe2x88x92xxc2x7(xcex24xe2x88x92xcex14)xe2x80x83xe2x80x83(28)
ipDC=DCxe2x80x83xe2x80x83(29)
Read Time correction, where xe2x80x98xcfx81xe2x80x99 is the channel read time of the diode array:
cxcex12=ipxcex12xe2x88x920.5xc2x7xcfx81xc2x7[(1+{square root over (2)})xc2x7ipxcex12+ipxcex22]xe2x80x83xe2x80x83(30)
cxcex22=ipxcex22xe2x88x920.5xc2x7xcfx81xc2x7[(1+{square root over (2)})xc2x7ipxcex22xe2x88x92ipxcex12]xe2x80x83xe2x80x83(31)
cxcex14=ipxcex14xe2x88x92xcfx81xc2x7(ipxcex14+ipxcex24)xe2x80x83xe2x80x83(32)
cxcex24=ipxcex24+xcfx81xc2x7(ipxcex14xe2x88x92ipxcex24)xe2x80x83xe2x80x83(33)
cDC=(1xe2x88x924xc2x7xcfx81/xcfx80)xc2x7ipDCxe2x80x83xe2x80x83(34)
Eqs. 25-34 can be applied after Eqs. 10-17 to account for non-idealities in the Photo Array Diode Element readout. The Image Persistence and Read-Out non-ideality factors xe2x80x98xxe2x80x99 and xe2x80x98xcfx81xe2x80x99 can also be determined by defining them as Fit Parameters in a Calibration Regression procedure presented in the following section of this Specification.
For demonstration purposes, considering now the present invention Spectroscopic Rotating Compensator Material System Investigation System to be a Rotating Compensator Ellipsometer System with Diode Element Array read-out, it must be understood that to acquire usable data, Calibration must be performed. Said calibration provides numerical values for Azimuthal Orientation Off-set Angles of Polarizer, Analyzer and Compensator with respect to a Material System Frame of Reference, along with the Retardance of the Rotating Compensator as a function of Wavelength. In addition, Calibration Parameters to compensate non-idealities in Diode Elements in a Photo Array are calibrated.
The foundation of the Calibration Procedure was first announced in the 1993 paper by Johs, published in Thin Film Solids, cited in the Background Section herein. The same basic Calibration Procedure technique is further developed in U.S. Pat. No. 5,706,212 which describes calibration of a Rotating Compensator Ellipsometer System utilized in the Infra-red (IR) band of wavelengths. Both identified references, however, describe typical application of the Regression based Calibration technique to one (1) wavelength at a time. While this method does work, it can require two-hundred-fifty-six (256) sets of Calibration Parameters where a two-hundred-fifty-six (256) Diode Element Photo Array is utilized, with each Diode Element serving to monitor an essentially single wavelength. (Note, as the electromagnetic spectrum is continuous, an essentially single wavelength is to be understood to be a small range of wavelengths centered around some wavelength, which essentially single wavelength is intercepted by a Diode Element in a Photo Array).
In practice of the present invention a xe2x80x9cGlobalxe2x80x9d regression procedure is typically performed on a Two (2) Dimensional Data Set. Typically Polarizer Azimuthal Angle and Wavelength are selected as Data Set Independent variables, although electromagnetic beam Angle-of-Incidence with respect to a Material System surface could be selected as an Independent variable instead of, for instance, Wavelength or Polarizer Azimuthal Angle. It is also noted that the Regression based Calibration described in U.S. Pat. No. 5,706,212 required that two (2), at least two (2) Dimensional Data Sets be provided in each Regression procedure. The two Data Sets are obtained with different investigated Material System configurations being employed. For instance, Data Sets utilizing two different Material Systems, or one Material System present and a xe2x80x9cStraight-throughxe2x80x9d configuration might be utilized. (Note, a xe2x80x9cStraight-throughxe2x80x9d configuration results when no Material System is present, and an electromagnetic beam is caused to pass sequentially through a Polarizer, Compensator and Analyzer then enter a Photo Array Detector System, without interacting with a Material System). The present invention, in its most basic embodiment, requires that only one Data Set be present. Said Data Set can be obtained with the Ellipsometer in Material System present or Straight-through configuration, although some benefits are realized when a Material System is utilized, (discussed supra herein). Of course, the present invention can be practiced utilizing multiple-Data Sets.
As mentioned, the Regression based Calibration procedure of the present invention requires that an at least Two (2) Dimensional Data Set be experimentally obtained. Typically said Two (2) Dimensional Data Set has as Independent Variables, Polarizer, (where the Rotating Compensator is placed after a Material System), Azimuthal Angle, and Wavelength. Where a Rotating Compensator is placed before a Material System, an Analyzer Azimuthal Angle is typically utilized. As mentioned, Angle-of-Incidence of an investigation Electromagnetic Beam with respect to an investigated Material system surface can be substituted for an Analyzer or Polarizer Azimuthal Angle settings, but this is not preferred as Material System PSI and DELTA values vary therewith. Also, it is generally simpler to vary a Polarizer or Analyzer Azimuthal Angle in most Ellipsometer systems in practice. Continuing, data is simultaneously obtained from many Diode Elements, (which correspond to different Wavelengths), and subjected to the Hadamard analysis inherent in Eqs. 20-24, infra, (and see also Eqs. 20xe2x80x2-24xe2x80x2 supra herein), to provide Fourier Coefficients present in Eq. 18. (It is noted that a Photo Array can contain 256, 1024 or 2048 Diode Elements, and some thereof might provide a signal which of too small an intensity to be utilized. The present invention allows for utilizing only a user selected group of signals for this and other reasons).
It will be noted that Eqs. 8 and 18 contain a D.C. term xe2x80x9cI0xe2x80x9d. This can be selected as a Fit Parameter in a Regression Procedure or a Normalization procedure can be implemented. Said Normalization can be with respect to the D.C. term, or a Normalizing Parameter can be included. The following Eqs 35a, 35b and 35c provide possible Normalizing Parameters:
Norm=DCxe2x80x83xe2x80x83(35a)
Norm={square root over ((xcex12+L )2+L +(xcex22+L )2+L +(xcex14+L )2+L +(xcex24+L )2+L +(DC)2+L )}xe2x80x83xe2x80x83(35b)
or
Norm={square root over ((xcex12+L )2+L +(xcex22+L )2+L +(xcex14+L )2+L +(xcex24+L )2+L )}xe2x80x83xe2x80x83(35c)
Eq. 35a provides for Normalizing with respect to the D.C. term, Eq. 35b provides for Normalizing to a Parameter which depends on the D.C. Term and the Fourier Coefficients, while Eq. 35c provides for Normalizing to a Parameter which depends on Fourier Coefficients but not the D.C. Term. If Fourier Coefficients are not Normalized, (ie. the D.C. Term xe2x80x9cI0xe2x80x9d is not included as a Fit Parameter in a Calibration Parameter evaluating Regression Procedure, or Normalization is not performed), it should be appreciated that a xe2x80x9cFloatingxe2x80x9d value result will be obtained for Calibration Parameters provided by application of the Calibration Parameter evaluating Regression onto said Fourier Series Coefficient values. As mentioned infra herein, the D.C. Component xe2x80x9cI0xe2x80x9d can be difficult to evaluate, often requiring a xe2x80x9cShutterxe2x80x9d to block background light, dark current, readout electronics voltage offsets etc. As well, the D.C. component is more susceptible to instrumentation drift. As a result, use of Eq. 35c can be preferrable in the present invention Calibration Procedure to use of Eqs. 35a and 35b and to including xe2x80x9cI0xe2x80x9d in a Regression Procedure for evaluating Calibration Parameters. (Note that calibration data is taken with the Rotating Compensator Sample System Investigating System in a xe2x80x9cSample Presentxe2x80x9d, rather than a xe2x80x9cStraight Throughxe2x80x9d configuration, where such Eq. 35c normalization is practiced). (Note, Eqs. 67, and accompanying discussion, provide additional insight to Calibration Normalization).
It is further noted that recent practice has adopted use of multiple data sets, similar to that described in U.S. Pat. No. 5,706,212 in present invention procedures to calibrate a present invention Rotating Compensator System, where it is desired to evaluate not only Ellipsometric Parameters, but Depolarization/Mueller Matrix values as well. Said multiple data sets can be obtained with different samples in place and/or with the ellipsometer system in a xe2x80x9cstraight-throughxe2x80x9d configuration. It is disclosed that it has been found desirable to normalize data to D.C. in some portions of calibration, and to an A.C. derived term in other portions thereof. An equation, such as presented in EQ. 35c, (which is derived from Fourier Coefficients), is an example of an A.C. data normalization parameter.
To shed light as to why various use of D.C. and A.C. based data normalization Parameters is beneficial, the following parameters are defined:
N=COS(2*PSI);
C=SIN(2*PSI)COS(DELTA);
and
xe2x80x83S=SIN(2*PSI)SIN(DELTA).
Further:
C=(fc(ALPHA 2, BETA2, ALPHA4, BETA4))/D.C.;
S=(fs(ALPHA 2, BETA2, ALPHA4, BETA4))/D.C.;
N=(fn(ALPHA 2, BETA2, ALPHA4, BETA4))/D.C.;
(where fc, fs and fn are functions to extract N, C and S from ALPHA2, BETA2, ALPHA4, BETA 4 AND D.C.); and
TAN(DELTA)=S/C, (note the D.C. term cancels);
TAN(PSI)=((C2+S2)(1/2))/N, (note the D.C.term cancels);
%DEPOL=100%(1xe2x88x92N2xe2x88x92C2xe2x88x92S2).
Thus it is demonstrated that PSI and DELTA can be calculated without the requirement of a D.C. term, but that calculation of Depolarization require knowledge of the D.C. term.
Preferred present invention calibration procedure practice provides that data be normalized to an A.C. derived basis, (eg. EQ. 35c), when determining such as compensator retardation (R), polarizer azimuth (P) and compensator fast axis azimuth (C), and that data be normalized to D.C., (eg. EQ. 35a or 35b), where optical element Depolarization/Meuller Matrix values are fit. Thus a calibration procedure as recited infra herein can be modified to include a step in which an appropriate normalization basis is determined at various steps therein.
(Note, in Eqs. 67, a different approach to defining Depolarization is presented. Nonideality Depolarization terms identified as xe2x80x98bxe2x80x99 and xe2x80x98cxe2x80x99 appear only in a D.C. term.) Where data normalization by D.C., or A.C, or a combination of D.C. and A.C. normalization data bases is practiced, the method of calibrating the spectroscopic rotating compensator material system investigation system can be recited as comprising, in any functional order, the steps of:
providing a spectroscopic rotating compensator material system investigation system comprising a source of a polychromatic beam of electromagnetic radiation, a polarizer, a stage for supporting a material system, an analyzer, a dispersive optics and at least one detector system which contains a multiplicity of detector elements, said spectroscopic rotating compensator material system investigation system further comprising at least one compensator(s) positioned at a location selected from the group consisting of: (before said stage for supporting a material system, and after said stage for supporting a material system, and both before and after said stage for supporting a material system); such that when said spectroscopic rotating compensator material system investigation system is used to investigate a material system present on said stage for supporting a material system, said analyzer and polarizer are maintained essentially fixed in position and at least one of said at least one compensator(s) is/are caused to continuously rotate while a polychromatic beam of electromagnetic radiation produced by said source of a polychromatic beam of electromagnetic radiation is caused to pass through said polarizer and said compensators), said polychromatic beam of electromagnetic radiation being also caused to interact with said material system, pass through said analyzer and interact with said dispersive optics such that a multiplicity of essentially single wavelengths are caused to simultaneously enter a corresponding multiplicity of detector elements in said at least one detector system;
at least one of said at least one compensator(s) preferably being a selection from the group consisting of:
comprised of a combination of at least two zero-order waveplates, said zero-order waveplates having their respective fast axes rotated to a position offset from zero or ninety degrees with respect to one another;
comprised of a combination of at least a first and a second effective zero-order wave plate, said first effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another, and said second effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another; the fast axes of the multiple order waveplates in said second effective zero-order wave plate being rotated to a position at a nominal forty-five degrees to the fast axes, respectively, of the multiple order waveplates in said first effective zero-order waveplate;
comprised of a combination of at least a first and a second effective zero-order wave plate, said first effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another, and said second effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another; the fast axes of the multiple order waveplates in said second effective zero-order wave plate being rotated to a position away from zero or ninety degrees with respect to the fast axes, respectively, of the multiple order waveplates in said first effective zero-order waveplate;
comprised a combination of at least one zero-order waveplate and at least one effective zero-order waveplate, said effective zero-order wave plate being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another, the fast axes of the multiple order waveplates in said effective zero-order wave plate being rotated to a position away from zero or ninety degrees with respect to the fast axis of the zero-order waveplate;
developing a mathematical model of said spectroscopic rotating compensator material system investigation system which comprises as calibration parameter variables such as polarizer azimuthal angle orientation, present material system PSI, present material system DELTA, compensator azimuthal angle orientation(s), matrix components of said compensator(s), and analyzer azimuthal angle orientation, which mathematical model is effectively a transfer function which enables calculation of electromagnetic beam magnitude as a function of wavelength detected by a detector element, given magnitude as a function of wavelength provided by said source of a polychromatic beam of electromagnetic radiation;
causing a polychromatic beam of electromagnetic radiation produced by said source of a polychromatic beam of electromagnetic radiation, to pass through said polarizer, interact with a material system caused to be in the path thereof, pass through said analyzer, and interact with said dispersive optics such that a multiplicity of essentially single wavelengths are caused to simultaneously enter a corresponding multiplicity of detector elements in said at least one detector system, with said polychromatic beam of electromagnetic radiation also being caused to pass through said compensator(s) positioned at a location selected from the group consisting of: (before said stage for supporting a material system, and after said stage for supporting a material system, and both before and after said stage for supporting a material system);
obtaining at least one, multi-dimensional, data set of magnitude values vs. wavelength and at least one parameter selected from the group consisting of:
angle-of-incidence of said polychromatic beam of electromagnetic radiation with respect to a present material system; and
azimuthal angle rotation of one element selected from the group consisting of:
said polarizer;
said analyzer;
OR
obtaining at least one multi-dimensional data set or at least two, at least one-dimensional, data sets of magnitude values vs. parameter(s) selected from the group consisting of:
wavelength;
angle-of-incidence of said polychromatic beam of electromagnetic radiation with respect to a present material system; and
azimuthal angle rotation of one element selected from the group consisting of:
said polarizer; and
said analyzer;
over time, while at least one of said at least one compensator(s) is caused to continuously rotate;
(It is noted here that a Reference Material System Thickness, or the Thickness of a Surface Layer thereupon, as well as DELTA Offset resulting from electromagentic beam passage through birefringent window(s) and/or lens(es), or wavelength shifts can be utilized as additional parameterization independent variables)
said data set(s) being obtained utilizing a selection from the group consiting of:
all of said data set(s), being obtained with a single material system placed on said stage for supporting a material system, with which material system said beam of electromagentic radiation is caused to interact;
at least one of said data set(s), being obtained utilizing one material system placed on said stage for supporting a material system, with another of said data set(s), being obtained utilizing another material system placed on said stage for supporting a material system, with which material system said beam of electromagentic radiation is caused to interact; and
at least one of said data set(s) being obtained with the spectroscopic rotating compensator material system investigation system oriented in a xe2x80x9cstraight-throughxe2x80x9d configuration wherein a polychromatic beam of electromagnetic radiation produced by said source of a polychromatic beam of electromagnetic radiation, is caused to pass through said polarizer, pass through said analyzer and interact with said dispersive optics such that a multiplicity of essentially single wavelengths are caused to simultaneously enter a corresponding multiplicity of detector elements in said at least one detector system, with said polychromatic beam of electromagnetic radiation also being caused to pass through at least one compensator(s) but without being caused to interact with any material system placed on said stage for supporting a material system other than open ambient atmosphere;
normalizing data in said data set(s) with respect to a selection from the group consisting of:
a data set D.C. component;
a data set A.C. component;
a parameter derived from a combinations of a data set D.C. component and a data set A.C. component;
performing a mathematical regression of said mathematical model onto said normalized data set(s) thereby evaluating calibration parameters in said mathematical model;
said regression based calibration procedure serving to evaluate parameters in said said mathematical model for non-achromatic characteristics and/or non-idealities and/or positions of at least one selection from the group consisting of:
azimuthal angle of said polarizer;
retardation of said compensator(s);
azimuthal angle(s) of said compensator(s), and
depolarization/Mueller Matrix components.
azimuthal angle of said analyzer.
optionally repeating STEPS E. and F. utilizing a different selection in STEP E. in normalizing data.
Continuing, normalized Fourier Coefficients can be represented by Eqs 36-39:                               n          ⁢                      xe2x80x83                    ⁢                      α            2                          =                              α            2                    Norm                                    (        36        )                                          n          ⁢                      xe2x80x83                    ⁢                      β            2                          =                              β            2                    Norm                                    (        37        )                                          n          ⁢                      xe2x80x83                    ⁢                      α            4                          =                              α            4                    Norm                                    (        38        )                                          n          ⁢                      xe2x80x83                    ⁢                      β            4                          =                              β            4                    Norm                                    (        39        )            
A Global Calibration Data Set can be represented by Eq. 40:
MFDP,n={(nxcex12)P,n, (nxcex22)P,n, (nxcex14)P,n, (nxcex24)P,n}xe2x80x83xe2x80x83(40)
where MFD stands for Measured Fourier Data, and where xe2x80x9cPxe2x80x9d is the Polarizer Angle and constitutes one Independent Variable, (and is typically varied within the range of from zero (0.0) to one-hundred-eighty (180) degrees, in ten (10) degree steps), and where xe2x80x9cnxe2x80x9d identifies the index of a selected Diode element, (channel), in the Photo Array, or alternatively stated, identifies a Second Independent Variable, (ie. Wavelength). It is noted that a typical system configuration would make use of Diode Elements (channels) 30-250 in a 256 channel Photo Array. The term xe2x80x9cGlobalxe2x80x9d emphasizes the presence of Wavelength Dependence. Utilizing the just described xe2x80x9cPxe2x80x9d range settings and Wavelength range, Eq. 41 indicates that the Global MFD Data Set would contain:
(180/10+1 polarizer settings)xc3x97(250xe2x88x9230+1 channels)xc3x97(4 Fourier components)=16,796xe2x80x83xe2x80x83(41)
It is further noted that an approximate error in Fourier Data can be estimated from signal to noise at each Detector Channel, and subsequently used in the Regression Analysis of the Experimentally Obtained Data Set.
Continuing, use of Eqs. 3-17, 35-39 and (25-34 if Photo Array non-idealities are included), allows one to calculate, (ie. mathematically predict), values of Normalized Fourier Coefficients as in Eqs 36-39, which will be experimentally measured by a present invention Rotating Compensator Material System Investigation System. However, to make said mathematical prediction requires that Material System PSI and DELTA values be known, the Offset Angles Ps, As, and Cs be known, and that Compensator Retardation xe2x80x9cxcex4xe2x80x9d be known as well as any other Compensator non-idealities, and that the Photo Array non-idealities xe2x80x9cx xe2x80x9d and xe2x80x9cxcfx81xe2x80x9d be known if necessary. Mathematically this can be represented by Eq. 42:
PFDP,n(P, xcexa8n, xcex94n, (Ps)n, (Cs)n, (As)n, xcex4n, xn, xcfx81n)xe2x80x83xe2x80x83(42)
Eq. 42 states that a Predicted Fourier Data (PFD) Set at a given Polarizer Azimuth and Photo Array Channel (Wavelength), is a function of identified variables, which variables constitute Calibration Parameters which must be provided numerical values. The present invention Regression procedure provides means for numerically evaluating the Calibration Parameters.
In all known prior art, separate Regression procedures have been carried out at each utilized Wavelength. If Two-Hundred (200) Wavelengths were utilized, then Two-Hundred (200) separate values for Ps, As and Cs etc. would be obtained. The present invention Regression Procedure, however, teaches that Calibration Parameters as a function of an Independent Variable, (eg. Wavelength), can be xe2x80x9cParameterizedxe2x80x9d. That is, a mathematical relationship requiring only a few (eg. perhaps two (2) or three (3) Parameters), can be generated to describe a functional relationship between the Calibration Parameter and the Independent Variable (eg. Wavelength), and the present invention Regression Procedure utilized to evaluate said Two (2) or Three (3) Parameters. For example, the Polarizer Azimuthal Offset (Ps) might be constant for all Wavelengths. Should this be the case then said Polarizer Azimuthal Offset (Ps) can be evaluated and stored, rather than, for instance, Two-Hundred (200) separate values at Two-Hundred (200) separate Wavelengths. In this instance, Eq. 43 indicates that a Global Calibration Parameter can be defined:
(Ps)nxe2x89xa1gPsxe2x80x83xe2x80x83(43)
In general, any of the discretely defined Calibration Parameters identified in Eq. 42, could be replaced by a Global Parametric Function as defined in Eq. 44:
CPn=gCP(n, p1, p2, . . . , pk)xe2x80x83xe2x80x83(44)
where CPn stands for any Calibration Parameter which is discretely defined for each xe2x80x9cnxe2x80x9d""th channel, (ie. the xe2x80x9cnxe2x80x9d""th Wavelength), and xe2x80x9cgCPxe2x80x9d is a global Parametric Function (as a function of an xe2x80x9cnxe2x80x9d""th channel number and xe2x80x9ckxe2x80x9d Calibration Parameters xe2x80x9cp1 . . . pk) which replace CPn. A Parametric Function can be of any mathematical form, such as, but not limited to, polynomial, rational or trancendental (in the case of xcexa8n and xcex94n, a Parametric Function could be calculated from a multi-layer optical model for a Material System, using known Material Optical Constants and Parameterized Film Thicknesses). The important characteristic of a Parametric Function being that:
1. It accurately represents the behavior of the Calibration Parameter at each Independent Variable (eg. Photo Array Channel or Wavelength).
2. It accurately represents the behavior of the Calibration Parameter utilizing fewer Parameters than would be required to simply evaluate Calibration Parameters at each utilized Independent Variable (eg. Wavelength).
In terms of Eq. 44 this can be stated that xe2x80x9ckxe2x80x9d (the number of Calibration Parameters), is less than xe2x80x9cnxe2x80x9d (the number of channels).
It is to be understood that preferred Global Parameter Function form utilized in the present invention depends upon the particular embodiment utilized, (eg. the Compensator type utilized). It is also within the scope of the present invention Regression based Calibration Parameter evaluation Procedure to represent some Calibration Parameters with Global Parametric Functions, and to represent other Calibration Parameters discretely. Three examples of Global Parametric Function utilizing Models follow directly.
Global Regression Mode (GRM) 1
This (GRM) requires that five (5) Calibration Parameters be evaluated. Eqs. 45-47 provide equations for Predicted Fourier Data (PFD):
PFDP,n(P, xcexa8n, xcex94n, gPs, gCs, gAs, gxcex4(n, p0, p1))xe2x80x83xe2x80x83(45)
where
gxcex4(n, p0, p1)=[p0xc2x790xc2x7(1+p1/[w(n)]2)]/w(n)xe2x80x83xe2x80x83(46)
and
w(n)=C0+C1xc2x7n+C2xc2x7n2xe2x80x83xe2x80x83(47)
where w(n) returns a wavelength of electromagnetic radiation (in nanometers), corresponding to the xe2x80x9cnxe2x80x9d""th channel of a Photo Array, where C0, C1 and C2 are wavelength Calibration Parameters. In the case where a previously identified Ziess Diode Array Spectrometer Systems is utilized, said C0, C1 and C2 Calibration Parameters are provided by the manufacturer, and Eq. 47 can be utilized to provide Wavelength given a Photo Array Channel number. The Global Retardance provided by a Compensator as a function of Wavelength is given by Eq. 46. Eq. 46 provides an Inverse Wavelength relationship, where xe2x80x9cp0xe2x80x9d is a Wavelength, (in nanometers), at which said Compensator is a xe2x80x9cQuarter-Wave-Platexe2x80x9d and demonstrates a Ninety (90) degree Retardation, and xe2x80x9cp1xe2x80x9d accounts for the Dispersive effects in the Optical Properties of the Compensator. Higher order terms can be added to Eq. 46.
In this (GRM) Mode 1, the Azimuthal Offset Calibration Parameters are considered constant for all Wavelengths. Therefore, using (GRM) Mode 1, only Five (5) Global Calibration Parameters:
(gPs, gCs, gAs, p0, p1)
in addition to Material System PSI and DELTA:
xcexa8n and xcex94n
need be evaluated by a present invention Regression Procedure.
Global Regression Mode (GRM) 2
This Mode is similar to (GRM) 1, but the Ps Calibration Parameter is defined as a Global Calibration Parameter, (ie. it is a constant independent of Photo Array Channel Number xe2x80x9cnxe2x80x9d) Again, the Retardance of the Compensator is Parameterized by Eqs. 46 and 47. Values for Cs and As are allowed to take on discrete vales at each Photo Array Channel, however, Eq. 48 indicates the relationship:
PFDP,n(P, xcexa8n, xcex94n, gPs, (Cs)n, (As)n, gxcex4(n, p0, p1))xe2x80x83xe2x80x83(48)
Global Refression Mode (GRM) 3
In this (GRM) 3 Mode, only Ps is defined as a Global Parameter, and all other system Calibration Parameters are allowed to take on discrete values at each Photo Array Channel. Eq. 49 indicates this relationship:
PFDP,n(P, xcexa8n, xcex94n, gPs, (Cs)n, (As)n, xcex4n)xe2x80x83xe2x80x83(49)
(SEE ALSO GLOBAL REGRESSION MODE (GRM) 4) SUPRA HEREIN.
Regression
Before turning to presentation of newly developed technique, it is noted that, as described in Parent U.S. Pat. No. 5,872,630, the present invention evaluates the Calibration Parameters identified Infra herein utilizing standard non-linear regression analysis. First a metric is defined by Eq. 50 to quantify Error between Calculated Predicted Fourier Data (PFD) and Experimentally Measured Fourier Data (MFD).                               χ          2                =                              ∑            P                    ⁢                                    ∑              n                        ⁢                                          (                                                                            MFD                                              P                        ,                        n                                                              -                                          PFD                      ⁡                                              (                                                  P                          ,                          n                          ,                                                      p                            k                                                                          )                                                                                                  σ                    ⁢                                          xe2x80x83                                        ⁢                                          MFD                                              P                        ,                        n                                                                                            )                            2                                                          (        50        )            
Eq. 50 is a simplified way of stating that overall error between measured and predicted Calibration Data Sets is given by the squared difference between each measured and corresponding calculated predicted Fourier data, normalized by the approximate error at each measured data point ("sgr"MFDP,n), and summed over all the Polarizer and Wavelength (Channel) setting values. Eq. 51 provides a more riggerous mathematical definition.                               χ          2                =                              ∑            P                    ⁢                                    ∑              n                        ⁢                          [                                                                                                                                            [                                                                                                                                                      (                                                                      m                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          α                                      2                                                                                                        )                                                                                                  P                                  ,                                  n                                                                                            -                                                              p                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  α                                                                      2                                    ⁢                                                                          (                                                                              P                                        ,                                        n                                        ,                                                                                  p                                          k                                                                                                                    )                                                                                                                                                                                                                                                          (                                                                  σ                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      α                                    2                                                                                                  )                                                                                            P                                ,                                n                                                                                                              ]                                                2                                            +                                                                                                    [                                                                                                                                                                (                                                                          m                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                                                              β                                        2                                                                                                              )                                                                                                        P                                    ,                                    n                                                                                                  -                                                                  p                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                                                            β                                      2                                                                        ⁡                                                                          (                                                                              P                                        ,                                        n                                        ,                                                                                  p                                          k                                                                                                                    )                                                                                                                                                                                                                                    (                                                                      σβ                                    2                                                                    )                                                                                                  P                                  ,                                  n                                                                                                                      ]                                                    2                                                ⁢                        ⋯                                            +                                                                                                                                                                                    [                                                                                                                                                      (                                                                      m                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          α                                      4                                                                                                        )                                                                                                  P                                  ,                                  n                                                                                            -                                                              p                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                  α                                                                      4                                    ⁢                                                                          (                                                                              P                                        ,                                        n                                        ,                                                                                  p                                          k                                                                                                                    )                                                                                                                                                                                                                                                          (                                                                  σ                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                                                      α                                    4                                                                                                  )                                                                                            P                                ,                                n                                                                                                              ]                                                2                                            +                                                                        [                                                                                                                                                      (                                                                      m                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                                                          β                                      4                                                                                                        )                                                                                                  P                                  ,                                  n                                                                                            -                                                              p                                ⁢                                                                  xe2x80x83                                                                ⁢                                                                                                      β                                    4                                                                    ⁡                                                                      (                                                                          P                                      ,                                      n                                      ,                                                                              p                                        k                                                                                                              )                                                                                                                                                                                                                      (                                                                  σβ                                  4                                                                )                                                                                            P                                ,                                n                                                                                                              ]                                                2                                                                                                        ]                                                          (        51        )            
In Eqs. 50 and 51, pk represents the xe2x80x9ckxe2x80x9d adjustable system Calibration Parameters required to calculate (PFD). The well known Marquardt-Levenberg non-linear Algorithm, as described in the Johs paper cited in the Background Section herein, can be used to itteratively adjust system Calibration Parameters pk to minimize error.
It is noted that good initial values are required to practice Regression which converges rapidly. The present invention obtains good starting values for use in the Global Regressions described, by performing a number of non-global Regressions at a multiplicity of discrete Wavelengths. The resulting ranges of values for the various Calibration Parameters then allows educated selection for Global Regression starting values.
It is also noted that Global Regression can be performed utilizing only data from every xe2x80x9cNxe2x80x9d""th Channel, (eg. every xe2x80x9cNxe2x80x9d""th Wavelength), to reduce required Regression procedure time to arrive at convergence. This approach to Regression is still to be considered as Global.
Once the present invention Spectroscopic Rotating Compensator Material System Investigation System is calibrated, it is possible to take data from unknown samples therewith and obtain PSI and DELTA plots therefore. Kleim et al., describes equations for PSI (xcexa8) and DELTA (xcex94) and these equations are provided as Eq. 52 and 53 herein:                               tan          ⁡                      (                          2              ·              Ψ                        )                          =                                                                              [                                                                                    (                                                  α                          2                                                )                                            2                                        +                                                                  (                                                  β                          2                                                )                                            2                                                        ]                                ·                                                      (                                                                  1                        -                                                  cos                          ⁡                                                      (                            δ                            )                                                                                                                      sin                        ⁡                                                  (                          δ                          )                                                                                      )                                    2                                            +                              4                ·                                                      (                                                                                            β                          4                                                ·                                                  cos                          ⁡                                                      (                                                          2                              ·                              P                                                        )                                                                                              -                                                                        α                          4                                                ·                                                  sin                          ⁡                                                      (                                                          2                              ·                              P                                                        )                                                                                                                )                                    2                                                                          2            ·                          (                                                                    α                    4                                    ·                                      cos                    ⁡                                          (                                              2                        ·                        P                                            )                                                                      +                                                      β                    4                                    ·                                      sin                    ⁡                                          (                                              2                        ·                        P                                            )                                                                                  )                                                          (        52        )                                          tan          ⁡                      (            Δ            )                          =                              (                                          1                -                                  cos                  ⁡                                      (                    δ                    )                                                                              2                ·                                  sin                  ⁡                                      (                    δ                    )                                                                        )                    ·                                                                      α                  2                                ·                                  sin                  ⁡                                      (                                          2                      ·                      P                                        )                                                              -                                                β                  2                                ·                                  cos                  ⁡                                      (                                          2                      ·                      P                                        )                                                                                                                        α                  4                                ·                                  sin                  ⁡                                      (                                          2                      ·                      P                                        )                                                              -                                                β                  4                                ·                                  cos                  ⁡                                      (                                          2                      ·                      P                                        )                                                                                                          (        53        )            
In these equations the Analyzer should be set to +/xe2x88x9245 degrees. Also, prior to applying Eqs. 52 and 53 the measured Fourier Data should be transformed into xe2x80x9cidealxe2x80x9d Fourier Data by application of Eqs. 15a, 15b, 16a, 16b, 17a and 17b as well as Eqs. 25-34. Kleim et al. also describes the advantages of performing a zone-averaged measurement in a Rotating Compensator System, (ie. averaging the PSI and DELTA extracted from measurements with the Analyzer A set to first, +45 Degrees, and second to xe2x88x9245 Degrees. This can be concurrently practiced with the present invention to further improve the accuracy of data measurement.
It is also noted that an alternative approach to obtaining Material System PSI and DELTA characterizing data, is to perform a Calibration Procedure on a present invention Spectroscopic Rotating Compensator Material System Investigation System in a Sample Present Mode, with said Material System present therein.
New Derivation of Frequency Components in General Rotating Compensator Ellipsometer/Polarimeters Sytems
Since the just reviewed teachings from Parent U.S. Pat. No. 5,872,630 were originally presented, additional work in the area has resulted in derivation of more generalized Equations which account for various frequency components which present in a Rotating Compensator ellipsometer or polarimeter system. Said derivation is based in application of general Mueller Matrix representations for optical elements. Eq. 54 provides said general Meuller Matrix representation:                     M        =                  [                                                    m11                                            m12                                            m13                                            m14                                                                    m21                                            m22                                            m23                                            m24                                                                    m31                                            m32                                            m33                                            m34                                                                    m41                                            m42                                            m43                                            m44                                              ]                            54      
Additionally, Eq. 55 demonstrates application of Rotation Matrices which serve to account for differences in angular orientation, (xe2x80x98xcfx86xe2x80x99) between a beam path coordinate system and an optical element coordinate system:                                                         Mrot              =                              xe2x80x83                            ⁢                                                [                                                                                    1                                                                    0                                                                    0                                                                    0                                                                                                            0                                                                                              cos                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                                                                                -                                                      sin                            ⁡                                                          (                                                              2                                ⁢                                φ                                                            )                                                                                                                                                  0                                                                                                            0                                                                                              sin                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                                                                                cos                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                                                      0                                                                                                            0                                                                    0                                                                    0                                                                    1                                                                              ]                                ·                                                                                                        xe2x80x83                            ⁢                                                                                                                                            [                                                                                                                    m11                                                                                            m12                                                                                            m13                                                                                            m14                                                                                                                                                    m21                                                                                            m22                                                                                            m23                                                                                            m24                                                                                                                                                    m31                                                                                            m32                                                                                            m33                                                                                            m34                                                                                                                                                    m41                                                                                            m42                                                                                            m43                                                                                            m44                                                                                                              ]                                                ·                                                                                            ⁡                                  [                                                                                    1                                                                    0                                                                    0                                                                    0                                                                                                            0                                                                                              cos                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                                                                                sin                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                                                      0                                                                                                            0                                                                                              -                                                      sin                            ⁡                                                          (                                                              2                                ⁢                                φ                                                            )                                                                                                                                                                            cos                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                                                      0                                                                                                            0                                                                    0                                                                    0                                                                    1                                                                              ]                                                                                55      
Multiplying Eq. 55 through provivdes:                     [                                            m11                                                                                            m12                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                -                                                      m13                    ·                    s2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                                                                                                          m12                    ·                    s2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                +                                                      m13                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                                                    m14                                                                                            c2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m21                                                  -                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m31                                                                                                                        c2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m22                                                  +                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m33                                                  -                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                                          (                                              m23                        +                        m32                                            )                                                                                                                                            c2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m23                                                  -                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m32                                                  +                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                                          (                                              m22                        -                        m33                                            )                                                                                                                                            c2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m24                                                  -                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m34                                                                                                                                            s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m21                                                  +                                  c2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m31                                                                                                                        c2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m32                                                  -                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m23                                                  +                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                                          (                                              m22                        -                        m33                                            )                                                                                                                                            c2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m33                                                  +                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                                            φ                      2                                        ·                    m22                                                  +                                  s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                                          (                                              m23                        +                        m32                                            )                                                                                                                                            s2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m24                                                  +                                  c2                  ⁢                                      xe2x80x83                                    ⁢                                      φ                    ·                    m34                                                                                                          m41                                                                                            m42                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                -                                                      m43                    ·                    s2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                                                                                                          m42                    ·                    s2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                +                                                      m43                    ·                    c2                                    ⁢                                      xe2x80x83                                    ⁢                  φ                                                                    m44                                      ]                    56      
where c2xcfx86=cos(2xcfx86) and s2xcfx86=sin(2xcfx86).
Now, if a general optical element is continuously rotated, the Matrix in Eq. 56 can be broken into a sum of matrices which 30 describe each frequency component, (ie. via Fourier Coefficients), and Eq. 57 demonstrates this:
Mrot=DC+A2xc2x7cos(2xcfx86)+B2xc2x7sin(2xcfx86)+A4xc2x7cos(4xcfx86)+B4xc2x7sin(4xcfx86)xe2x80x83xe2x80x8357
where:                               D          ⁢                      xe2x80x83                    ⁢          C                =                  [                                                    m11                                            0                                            0                                            m14                                                                    0                                                                                  m22                    +                    m33                                    2                                                                                                  m23                    -                    m32                                    2                                                            0                                                                    0                                                                                  m32                    -                    m23                                    2                                                                                                  m22                    +                    m33                                    2                                                            0                                                                    m41                                            0                                            0                                            m44                                              ]                            58                          A2        =                  [                                                    0                                            m12                                            m13                                            0                                                                    m21                                            0                                            0                                            m24                                                                    m31                                            0                                            0                                            m34                                                                    0                                            m42                                            m43                                            0                                              ]                                    xe2x80x83                                B2        =                  [                                                    0                                                              -                  m13                                                            m12                                            0                                                                                      -                  m31                                                            0                                            0                                                              -                  m34                                                                                    m21                                            0                                            0                                            m24                                                                    0                                                              -                  m43                                                            m42                                            0                                              ]                                    xe2x80x83                                A4        =                  [                                                    0                                            0                                            0                                            0                                                                    0                                                                                  m22                    -                    m33                                    2                                                                                                  m32                    +                    m23                                    2                                                            0                                                                    0                                                                                  m32                    +                    m23                                    2                                                                                                  m33                    -                    m22                                    2                                                            0                                                                    0                                            0                                            0                                            0                                              ]                                    xe2x80x83                                B4        =                  [                                                    0                                            0                                            0                                            0                                                                    0                                                                                                        -                      m32                                        -                    m23                                    2                                                                                                  m22                    -                    m33                                    2                                                            0                                                                    0                                                                                  m22                    -                    m33                                    2                                                                                                  m32                    +                    m23                                    2                                                            0                                                                    0                                            0                                            0                                            0                                              ]                                    xe2x80x83            
The frequency content for an arbitrary rotating optical element placed into an optical system can thus be easily calculated by inserting the appropriate frequency content matrix into the product of the Mueller Matracies which mathematically represent the optical system. And it is noted that while rotating a general optical element produces only D.C. and even harmonics, (ie. 2nd and 4th harmonics relative to the rotation frequency), if the Mueller Matrix elements are not constant as a function of rotation angle, additional xe2x80x9coddxe2x80x9d, and higher order harmonics could also be generated.
Proceeding, assuming that an ellipsometer system consists of an input polarizer with azimuthal angle P, a general rotating element, an analyzer with an azimuthal angle A, and a detector, an equation for Intensity output from a General Rotating Element Ellipsometer System can be derived as follows:                                                         I              =                              xe2x80x83                            ⁢                                                (                                                                                    1                                                                    0                                                                    0                                                                    0                                                                              )                                ·                                  [                                                                                    1                                                                    0                                                                    0                                                                    0                                                                                                            0                                                                                              cos                          ⁡                                                      (                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              A                                                        )                                                                                                                                                -                                                      sin                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                A                                                            )                                                                                                                                                  0                                                                                                            0                                                                                              sin                          ⁡                                                      (                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              A                                                        )                                                                                                                                                cos                          ⁡                                                      (                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              A                                                        )                                                                                                                      0                                                                                                            0                                                                    0                                                                    0                                                                    1                                                                              ]                                ·                                                                                                        xe2x80x83                            ⁢                                                [                                                                                    1                                                                    1                                                                    0                                                                    0                                                                                                            1                                                                    1                                                                    0                                                                    0                                                                                                            0                                                                    0                                                                    0                                                                    0                                                                                                            0                                                                    0                                                                    0                                                                    0                                                                              ]                                ·                                                                                                                              [                                                                                                                    1                                                                                            0                                                                                            0                                                                                            0                                                                                                                                                    0                                                                                                                              cos                                  ⁡                                                                      (                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      A                                                                        )                                                                                                                                                                                                sin                                  ⁡                                                                      (                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      A                                                                        )                                                                                                                                                              0                                                                                                                                                    0                                                                                                                              -                                                                      sin                                    ⁡                                                                          (                                                                              2                                        ⁢                                                                                  xe2x80x83                                                                                ⁢                                        A                                                                            )                                                                                                                                                                                                                                    cos                                  ⁡                                                                      (                                                                          2                                      ⁢                                                                              xe2x80x83                                                                            ⁢                                      A                                                                        )                                                                                                                                                              0                                                                                                                                                    0                                                                                            0                                                                                            0                                                                                            1                                                                                                              ]                                                ·                                                                                                                                                                                    xe2x80x83                            ⁢                                                [                                                                                    1                                                                                              -                          N                                                                                            0                                                                    0                                                                                                                                      -                          N                                                                                            1                                                                    0                                                                    0                                                                                                            0                                                                    0                                                                    C                                                                    S                                                                                                            0                                                                    0                                                                                              -                          S                                                                                            C                                                                              ]                                ·                Mrot                ·                                  [                                                                                    1                                                                                                                                      cos                          ⁡                                                      (                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              P                                                        )                                                                                                                                                                                        sin                          ⁡                                                      (                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              P                                                        )                                                                                                                                                              0                                                                              ]                                                                                59      
and simplifying yields:                                                         I              =                              xe2x80x83                            ⁢                                                (                                      1                    -                                                                  cos                        ⁡                                                  (                                                      2                            ·                            A                                                    )                                                                    ·                      N                                        -                    N                    +                                                                  cos                        ⁡                                                  (                                                      2                            ·                            A                                                    )                                                                    ⁢                                              xe2x80x83                                            ⁢                                                                        sin                          ⁡                                                      (                                                          2                              ·                              A                                                        )                                                                          ·                        C                                            ⁢                                              xe2x80x83                                            ⁢                                                                        sin                          ⁡                                                      (                                                          2                              ·                              A                                                        )                                                                          ·                        S                                                                              )                                ·                                                                                                        xe2x80x83                            ⁢                              Mrot                ·                                  [                                                                                    1                                                                                                                                      cos                          ⁡                                                      (                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              P                                                        )                                                                                                                                                                                        sin                          ⁡                                                      (                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              P                                                        )                                                                                                                                                              0                                                                              ]                                                                                60      
where an isotropic matrix was used to represent the material system sample.
To calculate the frequency content of the measured Intensity, the general rotating frequency component matracies are sequentially substituted for Mrot in Eq. 60 to yield:                                           D            ⁢                          xe2x80x83                        ⁢            C                    =                                    (                              1                -                                                      cos                    ⁡                                          (                                              2                        ·                        A                                            )                                                        ·                  N                                -                N                +                                                      cos                    ⁡                                          (                                              2                        ·                        A                                            )                                                        ⁢                                      xe2x80x83                                    ⁢                                                            sin                      ⁡                                              (                                                  2                          ·                          A                                                )                                                              ·                    C                                    ⁢                                      xe2x80x83                                    ⁢                                                            sin                      ⁡                                              (                                                  2                          ·                          A                                                )                                                              ·                    S                                                              )                        ·                                                                                                      [                                                                                                    m11                                                                                0                                                                                0                                                                                m14                                                                                                                                0                                                                                                                                              m22                                +                                m33                                                            2                                                                                                                                                                          m23                                -                                m32                                                            2                                                                                                            0                                                                                                                                0                                                                                                                                              m32                                -                                m23                                                            2                                                                                                                                                                          m22                                +                                m33                                                            2                                                                                                            0                                                                                                                                m41                                                                                0                                                                                0                                                                                m44                                                                                              ]                                        ·                                          [                                                                                                    1                                                                                                                                                              cos                              ⁡                                                              (                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  P                                                                )                                                                                                                                                                                                                        sin                              ⁡                                                              (                                                                  2                                  ⁢                                                                      xe2x80x83                                                                    ⁢                                  P                                                                )                                                                                                                                                                                          0                                                                                              ]                                                                                                          ⁢                  
                ⁢                                                                              A2                  =                                                            (                                              1                        -                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ·                          N                                                -                        N                        +                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            C                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            S                                                                                              )                                        ·                                                                                ⁢                                                                                          [                                                                                            0                                                                          m12                                                                          m13                                                                          0                                                                                                                      m21                                                                          0                                                                          0                                                                          m24                                                                                                                      m31                                                                          0                                                                          0                                                                          m34                                                                                                                      0                                                                          m42                                                                          m43                                                                          0                                                                                      ]                                    ·                                      [                                                                                            1                                                                                                                                                  cos                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                                                        sin                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                            0                                                                                      ]                                                                                      ⁢                  
                ⁢                                                                              B2                  =                                                            (                                              1                        -                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ·                          N                                                -                        N                        +                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            C                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            S                                                                                              )                                        ·                                                                                ⁢                                                                                          [                                                                                            0                                                                                                      -                            m13                                                                                                    m12                                                                          0                                                                                                                                                  -                            m31                                                                                                    0                                                                          0                                                                                                      -                            m34                                                                                                                                                m21                                                                          0                                                                          0                                                                          m24                                                                                                                      0                                                                                                      -                            m43                                                                                                    m42                                                                          0                                                                                      ]                                    ·                                      [                                                                                            1                                                                                                                                                  cos                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                                                        sin                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                            0                                                                                      ]                                                                                      ⁢                  
                ⁢                                                                              A4                  =                                                            (                                              1                        -                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ·                          N                                                -                        N                        +                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            C                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            S                                                                                              )                                        ·                                                                                ⁢                                                                                          [                                                                                            0                                                                          0                                                                          0                                                                          0                                                                                                                      0                                                                                                                                    m22                              -                              m33                                                        2                                                                                                                                                              m32                              +                              m23                                                        2                                                                                                    0                                                                                                                      0                                                                                                                                    m32                              +                              m23                                                        2                                                                                                                                                              m33                              -                              m22                                                        2                                                                                                    0                                                                                                                      0                                                                          0                                                                          0                                                                          0                                                                                      ]                                    ·                                      [                                                                                            1                                                                                                                                                  cos                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                                                        sin                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                            0                                                                                      ]                                                                                      ⁢                  
                ⁢                                                                              B4                  =                                                            (                                              1                        -                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ·                          N                                                -                        N                        +                                                                              cos                            ⁡                                                          (                                                              2                                ·                                A                                                            )                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            C                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                                                    sin                              ⁡                                                              (                                                                  2                                  ·                                  A                                                                )                                                                                      ·                            S                                                                                              )                                        ·                                                                                ⁢                                                                                          [                                                                                            0                                                                          0                                                                          0                                                                          0                                                                                                                      0                                                                                                                                                                    -                                m32                                                            -                              m23                                                        2                                                                                                                                                              m22                              -                              m33                                                        2                                                                                                    0                                                                                                                      0                                                                                                                                    m22                              -                              m33                                                        2                                                                                                                                                              m32                              -                              m23                                                        2                                                                                                    0                                                                                                                      0                                                                          0                                                                          0                                                                          0                                                                                      ]                                    ·                                      [                                                                                            1                                                                                                                                                  cos                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                                                        sin                            ⁡                                                          (                                                              2                                ⁢                                                                  xe2x80x83                                                                ⁢                                P                                                            )                                                                                                                                                                            0                                                                                      ]                                                                                                  61      
where
N=cos(2xcexa8), C=sin(2xcexa8)cos(xcex94), S=sin(2xcexa8)sin(xcex94),
C2A=cos(2A), S2A=sin(2A), C2P=cos(2P), S2P=sin(2P):
Final Fourier Coefficients for a general rotating element ellipsometer system, where xe2x80x98xe2x80x98xcfx86xe2x80x99xe2x80x99 is the rotating azimuth of the rotating element:                                                                                                                                                             Beam                        xe2x80x94                                            ⁢                      Intensity                                        =                                          xe2x80x83                                        ⁢                                                                  D                        ⁢                                                  xe2x80x83                                                ⁢                        C                                            +                                              A2                        ·                                                  cos                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                              +                                              B2                        ·                                                  sin                          ⁡                                                      (                                                          2                              ⁢                              φ                                                        )                                                                                              +                                                                                                                                                              xe2x80x83                                        ⁢                                                                  A4                        ·                                                  cos                          ⁡                                                      (                                                          4                              ⁢                              φ                                                        )                                                                                              +                                              B4                        ·                                                  sin                          ⁡                                                      (                                                          4                              ⁢                              φ                                                        )                                                                                                                                                                                                                                                                                                  D                      ⁢                                              xe2x80x83                                            ⁢                      C                                        =                                          xe2x80x83                                        ⁢                                                                  (                                                                                                            -                              m11                                                        ·                            C2A                                                    -                                                      C2P                            ·                                                          (                                                              m22                                +                                m33                                                            )                                                                                +                                                      S2P                            ·                                                          (                                                              m32                                -                                m23                                                            )                                                                                                      )                                            ·                                                                                                                                                              xe2x80x83                                        ⁢                                                                  N                        2                                            +                                                                        (                                                                                    C2P                              ·                              S2A                              ·                                                              (                                                                  m32                                  -                                  m23                                                                )                                                                                      +                                                          S2P                              ·                              S2A                              ·                                                              (                                                                  m22                                  +                                  m33                                                                )                                                                                                              )                                                ·                                                                                                                                                                                    xe2x80x83                                        ⁢                                                                                            C                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        …                                            +                                              S2A                        ·                        S                        ·                        m41                                            +                      m11                      +                                                                        1                          2                                                ·                        C2P                        ·                        C2A                        ·                                                                                                                                                                                    xe2x80x83                                        ⁢                                                                  (                                                  m22                          +                          m33                                                )                                            +                                                                        1                          2                                                ·                        S2P                        ·                        C2A                        ·                                                  (                                                      m23                            -                            m32                                                    )                                                                                                                                                                                                                                                  A2                    =                                          xe2x80x83                                        ⁢                                                                                            (                                                                                    -                              m21                                                        -                                                          C2A                              ·                                                              (                                                                                                      m12                                    ·                                    C2P                                                                    +                                                                      S2P                                    ·                                    m13                                                                                                  )                                                                                                              )                                                ·                        N                                            +                                                                                                                                                              xe2x80x83                                        ⁢                                                                                            (                                                      S2A                            ·                                                          (                                                                                                m42                                  ·                                  C2P                                                                +                                                                  S2P                                  ·                                  m43                                                                                            )                                                                                )                                                ·                        S                                            +                                              m21                        ·                        C2A                                            +                                                                                                                                                              xe2x80x83                                        ⁢                                                                  S2A                        ·                        C                        ·                        m31                                            +                                              C2P                        ·                        m12                                            +                                              S2P                        ·                        m                                                                                                                                                                                                                          B2                    =                                          xe2x80x83                                        ⁢                                                                                            (                                                      m31                            +                                                          C2A                              ·                                                              (                                                                                                      C2P                                    ·                                    m13                                                                    -                                                                      S2P                                    ·                                    m12                                                                                                  )                                                                                                              )                                                ·                        N                                            +                                                                                                                                                              xe2x80x83                                        ⁢                                                                                            (                                                                                    -                              S2A                                                        ·                                                          (                                                                                                C2P                                  ·                                  m43                                                                +                                                                  m42                                  ·                                  S2P                                                                                            )                                                                                )                                                ·                        S                                            -                                              m31                        ·                        C2A                                            +                                                                                                                                                              xe2x80x83                                        ⁢                                                                  S2A                        ·                        C                        ·                        m21                                            -                                              C2P                        ·                        m13                                            +                                              S2P                        ·                        m                                                                                                                                                                                                                          A4                    =                                          xe2x80x83                                        ⁢                                                                                            (                                                                                    C2P                              ·                                                              (                                                                  m33                                  -                                  m22                                                                )                                                                                      -                                                          S2P                              ·                                                              (                                                                  m32                                  +                                  m23                                                                )                                                                                                              )                                                ·                                                  N                          2                                                                    +                                                                                                                                                              xe2x80x83                                        ⁢                                                                  (                                                                              C2P                            ·                            S2A                            ·                                                          (                                                              m32                                +                                m23                                                            )                                                                                +                                                      S2P                            ·                            S2A                            ·                                                          (                                                              m33                                -                                m22                                                            )                                                                                                      )                                            ·                                                                                                                                                              xe2x80x83                                        ⁢                                                                                            C                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        …                                            +                                                                        1                          2                                                ·                        S2P                        ·                        C2A                        ·                                                  (                                                      m32                            +                            m23                                                    )                                                                    +                                                                        1                          2                                                ·                        C2P                        ·                        C2A                        ·                                                                                                                                                                                    xe2x80x83                                        ⁢                                          (                                              m22                        -                        m33                                            )                                                                                                                                                                                                    B4                    =                                          xe2x80x83                                        ⁢                                                                                            (                                                                                    C2P                              ·                                                              (                                                                  m32                                  +                                  m23                                                                )                                                                                      +                                                          S2P                              ·                                                              (                                                                  m33                                  -                                  m22                                                                )                                                                                                              )                                                ·                                                  N                          2                                                                    +                                                                                                                                                              xe2x80x83                                        ⁢                                                                  (                                                                              C2P                            ·                            S2A                            ·                                                          (                                                              m22                                -                                m33                                                            )                                                                                +                                                      S2P                            ·                            S2A                            ·                                                          (                                                              m32                                +                                m23                                                            )                                                                                                      )                                            ·                                                                                                                                                              xe2x80x83                                        ⁢                                                                                            C                          2                                                ⁢                                                  xe2x80x83                                                ⁢                        …                                            +                                                                        1                          2                                                ·                        S2P                        ·                        C2A                        ·                                                  (                                                      m22                            -                            m33                                                    )                                                                    -                                                                        1                          2                                                ·                        C2P                        ·                        C2A                        ·                                                                                                                                                                                    xe2x80x83                                        ⁢                                          (                                              m32                        +                        m23                                            )                                                                                                                      62      
If the general rotating element is an ideal compensator represented by the Mueller Matrix in Eq. 63, the Fourier Coefficients simplify to the expressions in Eq. 64.                                           M            xe2x80x94                    ⁢                      ideal            xe2x80x94                    ⁢          compensator                =                  [                                                    1                                            0                                            0                                            0                                                                    0                                            1                                            0                                            0                                                                    0                                            0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  δ                                                                              sin                  ⁢                                      xe2x80x83                                    ⁢                  δ                                                                                    0                                            0                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                  δ                                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                  δ                                                              ]                            63                                                                                                D                  ⁢                                      xe2x80x83                                    ⁢                  C                                =                                  xe2x80x83                                ⁢                                                      (                                                                  S2P                        ·                        S2A                        ·                        C                                            +                                              C2P                        ·                                                  (                                                      C2A                            -                            N                                                    )                                                                                      )                                    ·                                                                                                                          xe2x80x83                                ⁢                                                                            (                                              1                        +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                          δ                                                                    )                                        2                                    +                  1                  -                                                            1                      2                                        ·                    N                    ·                    C2A                                                                                      ⁢                  
                ⁢                  A2          =                                    S2A              ·              S2P              ·              sin                        ⁢                          xe2x80x83                        ⁢                          δ              ·              S                                      ⁢                  
                ⁢                  B2          =                                    S2A              ·              C2P              ·              sin                        ⁢                          xe2x80x83                        ⁢                          δ              ·              S                                      ⁢                  
                ⁢                  A4          =                                                    (                                  1                  -                                      cos                    ⁢                                          xe2x80x83                                        ⁢                    δ                                                  )                            2                        ·                          (                                                C2P                  ·                                      (                                          C2A                      -                      N                                        )                                                  -                                  S2A                  ·                  S2P                  ·                  C                                            )                                      ⁢                  
                ⁢                  B4          =                                                    (                                  1                  -                                      cos                    ⁢                                          xe2x80x83                                        ⁢                    δ                                                  )                            2                        ·                          (                                                S2P                  ·                                      (                                          C2A                      -                      N                                        )                                                  +                                  S2A                  ·                  C2P                  ·                  C                                            )                                                  64      
In actual rotating compensator systems, finite bandwidth and imperfect collimination can induce an apparent depolarization into the Mueller Matrix of the compensator of the form shown in Eq. 65:                                           M            xe2x80x94                    ⁢                      actual            xe2x80x94                    ⁢          compensator                =                  [                                                    1                                            0                                            0                                            0                                                                    0                                                              1                  -                  c                                                            0                                            0                                                                    0                                            0                                                              cos                  ⁢                                      xe2x80x83                                    ⁢                                      δ                    ·                                          (                                              1                        -                        b                                            )                                                                                                                    sin                  ⁢                                      xe2x80x83                                    ⁢                                      δ                    ·                                          (                                              1                        -                        b                                            )                                                                                                                          0                                            0                                                                                  -                    sin                                    ⁢                                      xe2x80x83                                    ⁢                                      δ                    ·                                          (                                              1                        -                        b                                            )                                                                                                                    cos                  ⁢                                      xe2x80x83                                    ⁢                                      δ                    ·                                          (                                              1                        -                        b                                            )                                                                                                    ]                            65      
and the Fourier Coeficients become:                                                                                           D                  ⁢                                      xe2x80x83                                    ⁢                  C                                =                                  xe2x80x83                                ⁢                                                      (                                                                  S2P                        ·                        S2A                        ·                        C                                            +                                              C2P                        ·                                                  (                                                      C2A                            -                            N                                                    )                                                                                      )                                    ·                                                                                                                          xe2x80x83                                ⁢                                                                            (                                              1                        +                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      δ                            ·                                                          (                                                              1                                -                                b                                                            )                                                                                                      -                        c                                            )                                        2                                    +                  1                  -                                                            1                      2                                        ·                    N                    ·                    C2A                                                                                      ⁢                  
                ⁢                  A2          =                                    S2A              ·              S2P              ·              sin                        ⁢                          xe2x80x83                        ⁢                          δ              ·              S              ·                              (                                  1                  -                  b                                )                                                    ⁢                  
                ⁢                  B2          =                                    S2A              ·              C2P              ·              sin                        ⁢                          xe2x80x83                        ⁢                          δ              ·              S              ·                              (                                  1                  -                  b                                )                                                    ⁢                  
                ⁢                                                            A4                =                                  xe2x80x83                                ⁢                                                                            (                                              1                        -                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      δ                            ·                                                          (                                                              1                                -                                b                                                            )                                                                                                      -                        c                                            )                                        2                                    ·                                                                                                                          xe2x80x83                                ⁢                                  (                                                            C2P                      ·                                              (                                                  C2A                          -                          N                                                )                                                              -                                          S2A                      ·                      S2P                      ·                      C                                                        )                                                                    ⁢                  
                ⁢                                                            B4                =                                  xe2x80x83                                ⁢                                                                            (                                              1                        -                                                  cos                          ⁢                                                      xe2x80x83                                                    ⁢                                                      δ                            ·                                                          (                                                              1                                -                                b                                                            )                                                                                                      -                        c                                            )                                        2                                    ·                                                                                                                          xe2x80x83                                ⁢                                  (                                                            S2P                      ·                                              (                                                  C2A                          -                          N                                                )                                                              +                                          S2A                      ·                      C2P                      ·                      C                                                        )                                                                                66      
If A.C. Normalization is used there is no sensitivity to these non-idealities, and the equations can be simply transformed as shown in Eqs. 67 to fit for an effective Compensator Retardance fxcex4t, and the effective non-idealities xe2x80x98bxe2x80x99 and xe2x80x98cxe2x80x99 will appear only in the D.C. term:                                                                                           D                  ⁢                                      xe2x80x83                                    ⁢                  C                                =                                  xe2x80x83                                ⁢                                  [                                                            (                                                                        S2P                          ·                          S2A                          ·                          C                                                +                                                  C2P                          ·                                                      (                                                          C2A                              -                              N                                                        )                                                                                              )                                        ·                                                                                                                        xe2x80x83                                                                                                                                                                            xe2x80x83                                                                                                                                                                                                                                                            xe2x80x83                                    ⁢                                                                                                              (                                                      1                            +                                                          cos                              ⁢                                                              xe2x80x83                                                            ⁢                              δ                                                        -                            c                                                    )                                                2                                            +                      1                      -                                                                        1                          2                                                ·                        N                        ·                        C2A                                                              ]                                                  ·                                  (                                      1                    +                    b                                    )                                                                    ⁢                  
                ⁢                  A2          =                                                    -                S2A                            ·              S2P              ·              sin                        ⁢                          xe2x80x83                        ⁢                          δ              ·              S                                      ⁢                  
                ⁢                  B2          =                                    S2A              ·              C2P              ·              sin                        ⁢                          xe2x80x83                        ⁢                          δ              ·              S                                      ⁢                  
                ⁢                  A4          =                                                    (                                  1                  -                                      cos                    ⁢                                          xe2x80x83                                        ⁢                    δ                                                  )                            2                        ·                          (                                                C2P                  ·                                      (                                          C2A                      -                      N                                        )                                                  -                                  S2A                  ·                  S2P                  ·                  C                                            )                                      ⁢                  
                ⁢                  B4          =                                                    (                                  1                  -                                      cos                    ⁢                                          xe2x80x83                                        ⁢                    δ                                                  )                            2                        ·                          (                                                S2P                  ·                                      (                                          C2A                      -                      N                                        )                                                  +                                  S2A                  ·                  C2P                  ·                  C                                            )                                                  67      
Inversion of Fourier Coefficients to Extract Ellipsometric Coefficients N, C and S
The preceeding Equations can be inverted to extract N, C and S, given the Fourier Coefficients that are measured by an ellipsometer system:                               N          =                                                                                          (                                                                                            (                                                                                    rDC                              ·                              C2P                              ·                              C2A                                                        +                            1                                                    )                                                ·                        a4                                            +                                              C2A                        ·                        b4                        ·                        S2P                        ·                        rDC                                            -                                              C2A                        ·                        cR                                                              )                                                                                      (                                                                    (                                                                  rDC                        ·                        C2P                                            +                      C2A                                        )                                    ·                  a4                                +                                  b4                  ·                  rDC                  ·                  S2P                                -                cR                            )                                      ⁢                  
                ⁢                  C          =                                    (                                                                    C2A                    2                                    -                  1                                S2A                            )                        ·                          b4                              (                                                                            (                                                                        rDC                          ·                          C2P                                                +                        C2A                                            )                                        ·                    a4                                    +                                      b4                    ·                    rDC                    ·                    S2P                                    -                  cR                                )                                                    ⁢                  
                ⁢                  S          =                                    (                                                                    C2A                    2                                    -                  1                                S2A                            )                        ·                          cR              sR                        ·                          a2                              (                                                                            (                                                                        rDC                          ·                          C2P                                                +                        C2A                                            )                                        ·                    a4                                    +                                      b4                    ·                    rDC                    ·                    S2P                                    -                  cR                                )                                                                68      
where       a2    =                                                      -              A2                        ·            S2P                    +                      B2            ·            C2P                                    D          ⁢                      xe2x80x83                    ⁢          C                    ·              (                  1          +          b                )                  a4    =                                        A4            ·            C2P                    +                      B4            ·            S2P                                    D          ⁢                      xe2x80x83                    ⁢          C                    ·              (                  1          +          b                )                  b4    =                                                      -              A4                        ·            S2P                    +                      B4            ·            C2P                                    D          ⁢                      xe2x80x83                    ⁢          C                    ·              (                  1          +          b                )                  rDC    =                                        (                          1              +                              cos                ⁢                                  xe2x80x83                                ⁢                δ                            -              c                        )                    2                ⁢                  xe2x80x83                ⁢        cR            =                                                  1              -                              cos                ⁢                                  xe2x80x83                                ⁢                δ                                      2                    ⁢                      xe2x80x83                    ⁢          sR                =                  sin          ⁢                      xe2x80x83                    ⁢          δ                    
The traditional Ellipsometric Parameters are given by:                               Δ          =                                    a              ⁢                              xe2x80x83                            ⁢                              tan                ⁡                                  (                                      S                    C                                    )                                            ⁢                              xe2x80x83                            ⁢              Ψ                        =                          a              ⁢                              xe2x80x83                            ⁢                              tan                ⁡                                  (                                                                                                              C                          2                                                +                                                  S                          2                                                                                      N                                    )                                                                    ⁢                  
                ⁢                                            %              xe2x80x94                        ⁢            Depolarization                    =                      100            ·                          (                              1                -                                  N                  2                                -                                  C                  2                                -                                  S                  2                                            )                                                  69      
and if the Analyzer Azimuth xe2x80x98Axe2x80x99 is set to forty-five (45) degrees, and they can be calculated without even measuring the D.C. component of the signal, although the D.C. component remains necessary to enable calculating Depolarization, (as in Eq. 70).                               Δ          =                      a            ⁢                          xe2x80x83                        ⁢                          tan              ⁡                              (                                                      cR                    sR                                    ·                                                                                    A2                        ·                        S2P                                            -                                              B2                        ·                        C2P                                                                                                            A4                        ·                        S2P                                            -                                              B4                        ·                        C2P                                                                                            )                                                    ⁢                  xe2x80x83                ⁢                  
                ⁢                  Ψ          =                      a            ⁢                          xe2x80x83                        ⁢                          tan              ⁡                              [                                                                                                                              (                                                                                                                    -                                A4                                                            ·                              S2P                                                        +                                                          B4                              ·                              C2P                                                                                )                                                2                                            +                                                                                                    (                                                          cR                              sR                                                        )                                                    2                                                ·                                                                              (                                                                                                                            -                                  A2                                                                ·                                S2P                                                            +                                                              B2                                ·                                C2P                                                                                      )                                                    2                                                                                                                                                A4                      ·                      C2P                                        +                                          B4                      ·                      S2P                                                                      ]                                                                70      
Where birefringent window(s) or lens(es) are present in the ellipsometric beam pathway and are characterized in terms of in-plane and out-of-plane retardance components, (as described in Allowed patent application Ser. No. 09/162,217, said 217 Application being incorporated hereinto by reference), the true N, C and S Parameters can be extracted from measured Fourier Coefficients using Eqs. 71. Note that out-of-plane window retardance effects on the data are analytically removed using these expressions:                               N          =                                                                                          (                                                                                            (                                                                                    sR                              ·                              cr1                                                        +                                                          sR                              ·                              rDC                              ·                              C2P                              ·                              C2A                              ·                              cr2                                                                                )                                                ·                        a4                                            -                                              cR                        ·                        a2                        ·                        sr1                                            +                                                                                                                                                                                      sR                        ·                        b4                        ·                        rDC                        ·                        S2P                        ·                        cr2                        ·                        C2A                                            -                                              sR                        ·                        cr2                        ·                        C2A                        ·                        cR                                                              )                                                                                                                                            (                                                                                            (                                                                                    sR                              ·                              C2P                              ·                              rDC                                                        +                                                          sR                              ·                              cr1                              ·                              cr2                              ·                              C2A                                                                                )                                                ·                        a4                                            -                                              sR                        ·                        cR                                            +                                                                                                                                                              S2P                      ·                      b4                      ·                      sR                      ·                      rDC                                        -                                                                  cR                        ·                        sr1                        ·                        a2                        ·                        C2A                        ·                        cr2                                            )                                                                                                          ⁢                  
                ⁢                  C          =                                                                                                                                                                  [                                                                                                                                                                                                                                                                        (                                                                                                                              -                                            sr2                                                                                    ·                                          cR                                          ·                                          C2A                                          ·                                                                                      (                                                                                                                                          cr2                                                ·                                                C2A                                                                                            +                                                                                              rDC                                                ·                                                C2P                                                ·                                                cr1                                                                                                                                      )                                                                                                                          )                                                                            ·                                      a2                                                                        ⁢                                                                          xe2x80x83                                                                        ⁢                                    …                                                                    +                                                                                                                                                                                                                      sR                                  ·                                                                      (                                                                                                                  C2A                                        ·                                        cr2                                        ·                                        cr1                                        ·                                        S2A                                                                            +                                                                              rDC                                        ·                                        C2P                                        ·                                        S2A                                                                            +                                                                                                                                                                                                                                                                                                                                                                                                              rDC                                        ·                                        S2P                                        ·                                        sr2                                        ·                                        C2A                                        ·                                        sr1                                                                            )                                                                        ·                                    b4                                                                    -                                                                      sR                                    ·                                    cR                                    ·                                    sr2                                    ·                                    C2A                                    ·                                    sr1                                                                                                                                                                                ]                                                ·                        N                                            ⁢                                              xe2x80x83                                            ⁢                      …                                        +                                                                                                                                                                  (                                                                              cR                            ·                            sr2                            ·                                                          C2A                              2                                                        ·                            rDC                            ·                            C2P                            ·                            cr1                            ·                            cr2                                                    +                                                      cR                            ·                            sr2                            ·                            C2A                                                                          )                                            ·                      a2                                        +                                                                                                                    (                                                                                            -                          sR                                                ·                        rDC                        ·                        C2P                        ·                        S2A                        ·                        C2A                        ·                        cr2                                            -                                              sR                        ·                        cr1                        ·                        S2A                                            -                                              sR                        ·                        rDC                        ·                        S2P                        ·                                                                                                                                                                                                                                                                  sr2                            ·                                                          C2A                              2                                                        ·                            sr1                            ·                            cr2                                                    )                                                ·                        b4                                            ⁢                                              xe2x80x83                                            ⁢                      …                                        +                                          cR                      ·                      sr2                      ·                                              C2A                        2                                            ·                      sR                      ·                      sr1                      ·                      cr2                                                                                                                                                                                      (                                                                        S2A                          2                                                +                                                                              C2A                            2                                                    ·                                                      sr2                            2                                                                                              )                                        ·                                                                                                                    (                                                                  sR                        ·                        b4                        ·                        rDC                        ·                        S2P                        ·                        cr1                                            +                                              cR                        ·                        a2                        ·                        rDC                        ·                        C2P                        ·                        sr1                                            -                                              sR                        ·                        cR                        ·                        cr1                                                              )                                                                                      ⁢                  
                ⁢                  S          =                                                                                                                                                                                                                                                            (                                                                                                                                            (                                                                                                                                                                    (                                                                                                                                          C2A                                                ·                                                cr2                                                                                            +                                                                                              rDC                                                ·                                                C2P                                                ·                                                cr1                                                                                                                                      )                                                                                    ·                                          a2                                                                                +                                                                                  sR                                          ·                                          sr1                                                                                                                    )                                                                        ·                                    N                                                                    +                                                                                                                                                                                                                                        (                                                                                                                                            (                                                                                                                        C2P                                          ·                                          sr1                                          ·                                          C2A                                          ·                                          sr2                                                                                -                                                                                  S2P                                          ·                                          S2A                                                                                                                    )                                                                        ·                                    a2                                    ·                                    rDC                                                                    -                                                                                                                                                                                                                                                                                                                                                                                                                                                  sR                                    ·                                    sr2                                    ·                                    cr1                                    ·                                    C2A                                                                    )                                                                ·                                C                                                            )                                                        ⁢                                                          xe2x80x83                                                        ⁢                            …                                                    +                                                      (                                                                                          (                                                                                                      -                                    1                                                                    -                                                                      rDC                                    ·                                    C2P                                    ·                                    cr1                                    ·                                    cr2                                    ·                                    C2A                                                                                                  )                                                            ·                                                                                                                                                                                                                                                                  a2                      -                                              sR                        ·                        sr1                        ·                        C2A                        ·                        cr2                                                              )                                                                                                                        (                                                            C2P                      ·                      sr1                      ·                      S2A                                        +                                          sr2                      ·                      C2A                      ·                      S2P                                                        )                                ·                a2                ·                rDC                            -                              sR                ·                cr1                ·                S2A                                                                71      
where
cr1=cos(r1), sr1=sin(r1), cr2=cos(r2), sr2=sin(r2), r1=out-of-plane entrance window retardance, r2=out-of-plane exit window retardance
It is noted that parameterization in calibration procedures can include DELTA offset due to birefringent effects of windows and/or lenses through which a beam of electromagentic radiation passes, as well as wavelength offsets, (eg. wherein a calculated curve is shifted along a wavelength axis while retaining it general shape by any means).
Global Regression Mode (GRM) 4
In this (GRM) 4 Mode Material System Sample PSI and DELTA are Parameterized as function of Reference Sample Surface Layer Thickness and Angle Of Incidence using well known optical model and optical constants for the substrate and film, and Compensator and Analyzer Characterizing Parameters are fit at each Wavelength. This serves to pick-up subtlties in Retardance, Fast Axis Position and Rotation. (GRM) 4 provides:
PFDPn(P, xcexa8(T, xcfx86), xcex94(T, xcfx86), gAs, (Cs)n, (Ps)n, xcex4n)xe2x80x83xe2x80x8372
provides that Reference Materials System PSI and DELTA parameters can be parameterized as functions of:
Reference Material System and or Surface Layer thereupon Thickness;
Angle of Incidence of the Electromagnetic Beam to a Reference Material System;
in addition to what is shown infra in the previously reported (GRM) 3 which is described by Eq. 49.
The present invention will be better understood by reference to the Detailed Description Section of this Disclosure, in conjunction with the accompanying Drawings.
It is therefore a primary objective and/or purpose of the present invention to teach a Spectroscopic Rotating Compensator Material System Investigation System, including at least one Photo Array comprised of a multiplicity of Diode Elements, for simultaneously detecting a Multiplicity of Wavelengths, which Spectroscopic Rotating Compensator Material System Investigation System can utilize both Achromatic and non-Achromatic Compensators of Berek-type with Optical Axis perpendicular to a surface thereof, and/or with Compensators with Optical Axis parallel to a surface thereof; and which Spectroscopic Rotating Compensator Material System Investigation System can be realized utilizing off-the-shelf Compensator and Spectrometer System components.
It is a further objective and/or purpose of the present ivnention to teach that a preferred Compensator Design is comprised of a combination of a first and a second actual or effective zero-order wave plate, said first actual or effective zero-order wave plate being either a single zero-order waveplate or being comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another, and said second actual or effective zero-order wave plate being either a single zero-order waveplate or comprised of two multiple order waveplates which are combined with the fast axes thereof oriented at a nominal ninety degrees to one another; the fast axes of the multiple order waveplate(s) in said second actual or effective zero-order wave plate being oriented at other than zero or ninety degrees, nominally forty-five degrees, to the fast axes, respectively, of the multiple order waveplate(s) in said first actual or effective zero-order waveplate.
It Is another primary objective and/or purpose of the present invention to teach, in the context of a Spectroscopic Rotating Compensator Material System Investigation System, Evaluation of Calibration Parameters in a Mathematical Model thereof by a Mathematical Regression based technique involving utilization of at least one, at least one dimensional data set, obtained with the Spectroscopic Rotating Compensator Material System Investigation System oriented in a xe2x80x9cMaterial System presentxe2x80x9d or in a xe2x80x9cStraight-throughxe2x80x9d configuration.
It is another objective and/or purpose of the present invention to teach, in the context of a Spectroscopic Rotating Compensator Material System Investigation System, Evaluation of Calibration Parameters in a Mathematical Model thereof by a Mathematical Regression based technique involving utilization of at least one multi-dimensional, data set(s) being obtained utilizing a selection from the group consiting of:
all of said at least one multi-dimensional data set(s), being obtained utilizing a single material system (MS) placed on said stage (STG) for supporting a material system (MS);
at least one of said at least one ulti-dimensional data set(s), being obtained utilizing one material system (MS) placed on said stage (STG) for supporting a material system (MS), with another of said at least one multi-dimensional data set(s), being obtained utilizing another material system (MS) placed on said stage (STG) for supporting a material system (MS); and
at least one of said at least one multi-dimensional data set(s) being obtained with the spectroscopic rotating compensator material system investigation system oriented in a xe2x80x9cstraight-throughxe2x80x9d configuration wherein a polychromatic beam of electromagnetic radiation (PPCLB) produced by said source (LS) of a polychromatic beam of electromagnetic radiation, is caused to pass through said polarizer (P), pass through said analyzer (A), and interact with said dispersive optics (DO) such that a multiplicity of essentially single wavelengths are caused to simultaneously enter a corresponding multiplicity of detector elements (DE""s) in said at least one detector system (DET), with said polychromatic beam of electromagnetic radiation (PPCLB) also being caused to pass through at least one compensator(s) (C) (Cxe2x80x2) (Cxe2x80x3) but without being caused to interact with any material system (MS) placed on said stage (STG) for supporting a material system (MS) other than open ambient atmosphere.
It is yet another objective and/or purpose of the present invention to teach, in the context of a Spectroscopic Rotating Compensator Material System Investigation System, Evaluation of Calibration Parameters in a Mathematical Model thereof by a Mathematical Regression based technique involving utilization of said at least two, at least one-dimensional, data set(s) being obtained utilizing a selection from the group consiting of:
all of said at least two, at least one-dimensional data set(s), being obtained utilizing a single material system (MS) placed on said stage (STG) for supporting a material system (MS);
at least one of said at least two, at least one-dimensional data set(s) being obtained utilizing one material system (MS) placed on said stage (STG) for supporting a material system (MS), and at least one of said at least two at least one-dimensional data set(s) being obtained utilizing one material system (MS) placed on said stage (STG) for supporting a material system (MS); and
at least one of said at least two, at least one-dimensional data set(s) being obtained utilizing one material system (MS) placed on said stage (STG) for supporting a material system (MS), and at least one of said at least two, at least one-dimensional data set(s) being obtained with the spectroscopic rotating compensator material system investigation system oriented in a xe2x80x9cstraight-throughxe2x80x9d configuration wherein a polychromatic beam of electromagnetic radiation (PPCLB) produced by said source (LS) of a polychromatic beam of electromagnetic radiation, is caused to pass through said polarizer (P), pass through said analyzer (A), and interact with said dispersive optics (DO) such that a multiplicity of essentially single wavelengths are caused to simultaneously enter a corresponding multiplicity of detector elements (DE""s) in said at least one detector system (DET), with said polychromatic beam of electromagnetic radiation (PPCLB) also being caused to pass through at least one compensator(s) (C) (Cxe2x80x2) (Cxe2x80x3) but without being caused to interact with any material system (MS) placed on said stage (STG) for supporting a material system (MS) other than open ambient atmosphere.
It is yet another objective and/or purpose of the present invention to teach that, where beneficial and desirable, Parameterization of Calibration Parameters, (such as Azimuthal Orientation Angle of Polarizer, Compensator(s) and Analyzer, and Material System PSI and DELTA, and Compensator Representing Matrix Components), as a function of a Data Set variable, (such as Wavelength, or Polarizer and/or Analyzer Azimuthal Angle Rotation, or Angle-of-Incidence of an electromagnetic beam with respect to a surface of a Material System being investigated, or Thickness of a Material System or Surface Layer thereupon, or a DELTA Offset resulting from passage of the electromagentic beam through a Birefringent Window or Lens, or a Wavelength Shift from a calculated ideal etc.), to reduce the number of Calibration Parameters which need be evaluated during a mathematical regression based Calibration Procedure, should be practiced.
It is another objective and/or purpose yet of the present invention, to teach, in the context of a Spectroscopic Rotating Compensator Material System Investigation System, Evaluation of Calibration Parameters in a Mathematical Model thereof by a Mathematical Regression based technique involving utilization of data set(s) which are normalized utilizing a selection from the group consisting of:
a data set D.C. component;
a data set A.C. component;
a parameter derived from a combinations of a data set D.C. component and a data set A.C. component;
It is a general objective and/or purpose of the present Disclosure to provide experimentally determined documentation of the utility of the present invention Spectroscopic Rotating Compensator Material System Investigation System, in the form of results obtained from practice of the Mathematical Regression Calibration Method, and the Material System Investigation Data Acquisition Method.
Other objectives and/or purposes will become obvious by a reading of the Specification.